Theory Hiding_Type_Variables
section‹Hiding Type Variables›
text‹ This theory\footnote{This theory can be used ``stand-alone,'' i.e., this theory is
not specific to the DOM formalization. The latest version is part of the ``Isabelle Hacks''
repository: \url{https://git.logicalhacking.com/adbrucker/isabelle-hacks/}.} implements
a mechanism for declaring default type variables for data types. This comes handy for complex
data types with many type variables.›
theory
"Hiding_Type_Variables"
imports
Main
keywords
"register_default_tvars"
"update_default_tvars_mode"::thy_decl
begin
section‹Implementation›
subsection‹Theory Managed Data Structure›
ML‹
signature HIDE_TVAR = sig
datatype print_mode = print_all | print | noprint
datatype tvar_subst = right | left
datatype parse_mode = parse | noparse
type hide_varT = {
name: string,
tvars: typ list,
typ_syn_tab : (string * typ list*string) Symtab.table,
print_mode: print_mode,
parse_mode: parse_mode
}
val parse_print_mode : string -> print_mode
val parse_parse_mode : string -> parse_mode
val register : string -> print_mode option -> parse_mode option ->
theory -> theory
val update_mode : string -> print_mode option -> parse_mode option ->
theory -> theory
val lookup : theory -> string -> hide_varT option
val hide_tvar_tr' : string -> Proof.context -> term list -> term
val hide_tvar_ast_tr : Proof.context -> Ast.ast list -> Ast.ast
val hide_tvar_subst_ast_tr : tvar_subst -> Proof.context -> Ast.ast list
-> Ast.ast
val hide_tvar_subst_return_ast_tr : tvar_subst -> Proof.context
-> Ast.ast list -> Ast.ast
end
structure Hide_Tvar : HIDE_TVAR = struct
datatype print_mode = print_all | print | noprint
datatype tvar_subst = right | left
datatype parse_mode = parse | noparse
type hide_varT = {
name: string,
tvars: typ list,
typ_syn_tab : (string * typ list*string) Symtab.table,
print_mode: print_mode,
parse_mode: parse_mode
}
type hide_tvar_tab = (hide_varT) Symtab.table
fun hide_tvar_eq (a, a') = (#name a) = (#name a')
fun merge_tvar_tab (tab,tab') = Symtab.merge hide_tvar_eq (tab,tab')
structure Data = Generic_Data
(
type T = hide_tvar_tab
val empty = Symtab.empty:hide_tvar_tab
val extend = I
fun merge(t1,t2) = merge_tvar_tab (t1, t2)
);
fun parse_print_mode "print_all" = print_all
| parse_print_mode "print" = print
| parse_print_mode "noprint" = noprint
| parse_print_mode s = error("Print mode not supported: "^s)
fun parse_parse_mode "parse" = parse
| parse_parse_mode "noparse" = noparse
| parse_parse_mode s = error("Parse mode not supported: "^s)
fun update_mode typ_str print_mode parse_mode thy =
let
val ctx = Toplevel.context_of(Toplevel.theory_toplevel thy)
val typ = Syntax.parse_typ ctx typ_str
val name = case typ of
Type(name,_) => name
| _ => error("Complex type not (yet) supported.")
fun update tab =
let
val old_entry = (case Symtab.lookup tab name of
SOME t => t
| NONE => error ("Type shorthand not registered: "^name))
val print_m = case print_mode of
SOME m => m
| NONE => #print_mode old_entry
val parse_m = case parse_mode of
SOME m => m
| NONE => #parse_mode old_entry
val entry = {
name = name,
tvars = #tvars old_entry,
typ_syn_tab = #typ_syn_tab old_entry,
print_mode = print_m,
parse_mode = parse_m
}
in
Symtab.update (name,entry) tab
end
in
Context.theory_of ( (Data.map update) (Context.Theory thy))
end
fun lookup thy name =
let
val tab = (Data.get o Context.Theory) thy
in
Symtab.lookup tab name
end
fun obtain_normalized_vname lookup_table vname =
case List.find (fn e => fst e = vname) lookup_table of
SOME (_,idx) => (lookup_table, Int.toString idx)
| NONE => let
fun max_idx [] = 0
| max_idx ((_,idx)::lt) = Int.max(idx,max_idx lt)
val idx = (max_idx lookup_table ) + 1
in
((vname,idx)::lookup_table, Int.toString idx) end
fun normalize_typvar_type lt (Type (a, Ts)) =
let
fun switch (a,b) = (b,a)
val (Ts', lt') = fold_map (fn t => fn lt => switch (normalize_typvar_type lt t)) Ts lt
in
(lt', Type (a, Ts'))
end
| normalize_typvar_type lt (TFree (vname, S)) =
let
val (lt, vname) = obtain_normalized_vname lt (vname)
in
(lt, TFree( vname, S))
end
| normalize_typvar_type lt (TVar (xi, S)) =
let
val (lt, vname) = obtain_normalized_vname lt (Term.string_of_vname xi)
in
(lt, TFree( vname, S))
end
fun normalize_typvar_type' t = snd ( normalize_typvar_type [] t)
fun mk_p s = s
fun key_of_type (Type(a, TS)) = mk_p (a^String.concat(map key_of_type TS))
| key_of_type (TFree (vname, _)) = mk_p vname
| key_of_type (TVar (xi, _ )) = mk_p (Term.string_of_vname xi)
val key_of_type' = key_of_type o normalize_typvar_type'
fun normalize_typvar_term lt (Const (a, t)) = (lt, Const(a, t))
| normalize_typvar_term lt (Free (a, t)) = let
val (lt, vname) = obtain_normalized_vname lt a
in
(lt, Free(vname,t))
end
| normalize_typvar_term lt (Var (xi, t)) =
let
val (lt, vname) = obtain_normalized_vname lt (Term.string_of_vname xi)
in
(lt, Free(vname,t))
end
| normalize_typvar_term lt (Bound (i)) = (lt, Bound(i))
| normalize_typvar_term lt (Abs(s,ty,tr)) =
let
val (lt,tr) = normalize_typvar_term lt tr
in
(lt, Abs(s,ty,tr))
end
| normalize_typvar_term lt (t1$t2) =
let
val (lt,t1) = normalize_typvar_term lt t1
val (lt,t2) = normalize_typvar_term lt t2
in
(lt, t1$t2)
end
fun normalize_typvar_term' t = snd(normalize_typvar_term [] t)
fun key_of_term (Const(s,_)) = if String.isPrefix "\<^type>" s
then Lexicon.unmark_type s
else ""
| key_of_term (Free(s,_)) = s
| key_of_term (Var(xi,_)) = Term.string_of_vname xi
| key_of_term (Bound(_)) = error("Bound() not supported in key_of_term")
| key_of_term (Abs(_,_,_)) = error("Abs() not supported in key_of_term")
| key_of_term (t1$t2) = (key_of_term t1)^(key_of_term t2)
val key_of_term' = key_of_term o normalize_typvar_term'
fun hide_tvar_tr' tname ctx terms =
let
val mtyp = Syntax.parse_typ ctx tname
val (fq_name, _) = case mtyp of
Type(s,ts) => (s,ts)
| _ => error("Complex type not (yet) supported.")
val local_name_of = hd o rev o String.fields (fn c => c = #".")
fun hide_type tname = Syntax.const("(_) "^tname)
val reg_type_as_term = Term.list_comb(Const(Lexicon.mark_type tname,dummyT),terms)
val key = key_of_term' reg_type_as_term
val actual_tvars_key = key_of_term reg_type_as_term
in
case lookup (Proof_Context.theory_of ctx) fq_name of
NONE => raise Match
| SOME e => let
val (tname,default_tvars_key) =
case Symtab.lookup (#typ_syn_tab e) key of
NONE => (local_name_of tname, "")
| SOME (s,_,tv) => (local_name_of s,tv)
in
case (#print_mode e) of
print_all => hide_type tname
| print => if default_tvars_key=actual_tvars_key
then hide_type tname
else raise Match
| noprint => raise Match
end
end
fun hide_tvar_ast_tr ctx ast=
let
val thy = Proof_Context.theory_of ctx
fun parse_ast ((Ast.Constant const)::[]) = (const,NONE)
| parse_ast ((Ast.Constant sort)::(Ast.Constant const)::[])
= (const,SOME sort)
| parse_ast _ = error("AST type not supported.")
val (decorated_name, decorated_sort) = parse_ast ast
val name = Lexicon.unmark_type decorated_name
val default_info = case lookup thy name of
NONE => error("No default type vars registered: "^name)
| SOME e => e
val _ = if #parse_mode default_info = noparse
then error("Default type vars disabled (option noparse): "^name)
else ()
fun name_of_tvar tvar = case tvar of (TFree(n,_)) => n
| _ => error("Unsupported type structure.")
val type_vars_ast =
let fun mk_tvar n =
case decorated_sort of
NONE => Ast.Variable(name_of_tvar n)
| SOME sort => Ast.Appl([Ast.Constant("_ofsort"),
Ast.Variable(name_of_tvar n),
Ast.Constant(sort)])
in
map mk_tvar (#tvars default_info)
end
in
Ast.Appl ((Ast.Constant decorated_name)::type_vars_ast)
end
fun register typ_str print_mode parse_mode thy =
let
val ctx = Toplevel.context_of(Toplevel.theory_toplevel thy)
val typ = Syntax.parse_typ ctx typ_str
val (name,tvars) = case typ of Type(name,tvars) => (name,tvars)
| _ => error("Unsupported type structure.")
val base_typ = Syntax.read_typ ctx typ_str
val (base_name,base_tvars) = case base_typ of Type(name,tvars) => (name,tvars)
| _ => error("Unsupported type structure.")
val base_key = key_of_type' base_typ
val base_tvar_key = key_of_type base_typ
val print_m = case print_mode of
SOME m => m
| NONE => print_all
val parse_m = case parse_mode of
SOME m => m
| NONE => parse
val entry = {
name = name,
tvars = tvars,
typ_syn_tab = Symtab.empty:((string * typ list * string) Symtab.table),
print_mode = print_m,
parse_mode = parse_m
}
val base_entry = if name = base_name
then
{
name = "",
tvars = [],
typ_syn_tab = Symtab.empty:((string * typ list * string) Symtab.table),
print_mode = noprint,
parse_mode = noparse
}
else case lookup thy base_name of
SOME e => e
| NONE => error ("No entry found for "^base_name^
" (via "^name^")")
val base_entry = {
name = #name base_entry,
tvars = #tvars base_entry,
typ_syn_tab = Symtab.update (base_key, (name, base_tvars, base_tvar_key))
(#typ_syn_tab (base_entry)),
print_mode = #print_mode base_entry,
parse_mode = #parse_mode base_entry
}
fun reg tab = let
val tab = Symtab.update_new(name, entry) tab
val tab = if name = base_name
then tab
else Symtab.update(base_name, base_entry) tab
in
tab
end
val thy = Sign.print_translation
[(Lexicon.mark_type name, hide_tvar_tr' name)] thy
in
Context.theory_of ( (Data.map reg) (Context.Theory thy))
handle Symtab.DUP _ => error("Type shorthand already registered: "^name)
end
fun hide_tvar_subst_ast_tr hole ctx (ast::[]) =
let
val thy = Proof_Context.theory_of ctx
val (decorated_name, args) = case ast
of (Ast.Appl ((Ast.Constant s)::args)) => (s, args)
| _ => error "Error in obtaining type constructor."
val name = Lexicon.unmark_type decorated_name
val default_info = case lookup thy name of
NONE => error("No default type vars registered: "^name)
| SOME e => e
val _ = if #parse_mode default_info = noparse
then error("Default type vars disabled (option noparse): "^name)
else ()
fun name_of_tvar tvar = case tvar of (TFree(n,_)) => n
| _ => error("Unsupported type structure.")
val type_vars_ast = map (fn n => Ast.Variable(name_of_tvar n)) (#tvars default_info)
val type_vars_ast = case hole of
right => (List.rev(List.drop(List.rev type_vars_ast, List.length args)))@args
| left => args@List.drop(type_vars_ast, List.length args)
in
Ast.Appl ((Ast.Constant decorated_name)::type_vars_ast)
end
| hide_tvar_subst_ast_tr _ _ _ = error("hide_tvar_subst_ast_tr: empty AST.")
fun hide_tvar_subst_return_ast_tr hole ctx (retval::constructor::[]) =
hide_tvar_subst_ast_tr hole ctx [Ast.Appl (constructor::retval::[])]
| hide_tvar_subst_return_ast_tr _ _ _ =
error("hide_tvar_subst_return_ast_tr: error in parsing AST")
end
›
subsection‹Register Parse Translations›
syntax "_tvars_wildcard" :: "type ⇒ type" ("'('_') _")
syntax "_tvars_wildcard_retval" :: "type ⇒ type ⇒ type" ("'('_, _') _")
syntax "_tvars_wildcard_sort" :: "sort ⇒ type ⇒ type" ("'('_::_') _")
syntax "_tvars_wildcard_right" :: "type ⇒ type" ("_ '_..")
syntax "_tvars_wildcard_left" :: "type ⇒ type" ("_ ..'_")
parse_ast_translation‹
[
(@{syntax_const "_tvars_wildcard_sort"}, Hide_Tvar.hide_tvar_ast_tr),
(@{syntax_const "_tvars_wildcard"}, Hide_Tvar.hide_tvar_ast_tr),
(@{syntax_const "_tvars_wildcard_retval"}, Hide_Tvar.hide_tvar_subst_return_ast_tr Hide_Tvar.right),
(@{syntax_const "_tvars_wildcard_right"}, Hide_Tvar.hide_tvar_subst_ast_tr Hide_Tvar.right),
(@{syntax_const "_tvars_wildcard_left"}, Hide_Tvar.hide_tvar_subst_ast_tr Hide_Tvar.left)
]
›
subsection‹Register Top-Level Isar Commands›
ML‹
val modeP = (Parse.$$$ "("
|-- (Parse.name --| Parse.$$$ ","
-- Parse.name --|
Parse.$$$ ")"))
val typ_modeP = Parse.typ -- (Scan.optional modeP ("print_all","parse"))
val _ = Outer_Syntax.command @{command_keyword "register_default_tvars"}
"Register default variables (and hiding mechanims) for a type."
(typ_modeP >> (fn (typ,(print_m,parse_m)) =>
(Toplevel.theory
(Hide_Tvar.register typ
(SOME (Hide_Tvar.parse_print_mode print_m))
(SOME (Hide_Tvar.parse_parse_mode parse_m))))));
val _ = Outer_Syntax.command @{command_keyword "update_default_tvars_mode"}
"Update print and/or parse mode or the default type variables for a certain type."
(typ_modeP >> (fn (typ,(print_m,parse_m)) =>
(Toplevel.theory
(Hide_Tvar.update_mode typ
(SOME (Hide_Tvar.parse_print_mode print_m))
(SOME (Hide_Tvar.parse_parse_mode parse_m))))));
›
subsection‹Introduction›
text‹
When modelling object-oriented data models in HOL with the goal of preserving ∗‹extensibility›
(e.g., as described in~\cite{brucker.ea:extensible:2008-b,brucker:interactive:2007}) one needs
to define type constructors with a large number of type variables. This can reduce the readability
of the overall formalization. Thus, we use a short-hand notation in cases were the names of
the type variables are known from the context. In more detail, this theory sets up both
configurable print and parse translations that allows for replacing @{emph ‹all›} type variables
by ‹(_)›, e.g., a five-ary constructor ‹('a, 'b, 'c, 'd, 'e) hide_tvar_foo› can
be shorted to ‹(_) hide_tvar_foo›. The use of this shorthand in output (printing) and
input (parsing) is, on a per-type basis, user-configurable using the top-level commands
‹register_default_tvars› (for registering the names of the default type variables and
the print/parse mode) and ‹update_default_tvars_mode› (for changing the print/parse mode
dynamically).
The input also supports short-hands for declaring default sorts (e.g., ‹(_::linorder)›
specifies that all default variables need to be instances of the sort (type class)
@{class ‹linorder›} and short-hands of overriding a suffice (or prefix) of the default type
variables. For example, ‹('state) hide_tvar_foo _.› is a short-hand for
‹('a, 'b, 'c, 'd, 'state) hide_tvar_foo›. In this document, we omit the implementation
details (we refer the interested reader to theory file) and continue directly with a few
examples.
›
subsection‹Example›
text‹Given the following type definition:›
datatype ('a, 'b) hide_tvar_foobar = hide_tvar_foo 'a | hide_tvar_bar 'b
type_synonym ('a, 'b, 'c, 'd) hide_tvar_baz = "('a+'b, 'a × 'b) hide_tvar_foobar"
text‹We can register default values for the type variables for the abstract
data type as well as the type synonym:›
register_default_tvars "('alpha, 'beta) hide_tvar_foobar" (print_all,parse)
register_default_tvars "('alpha, 'beta, 'gamma, 'delta) hide_tvar_baz" (print_all,parse)
text‹This allows us to write›
definition hide_tvar_f::"(_) hide_tvar_foobar ⇒ (_) hide_tvar_foobar ⇒ (_) hide_tvar_foobar"
where "hide_tvar_f a b = a"
definition hide_tvar_g::"(_) hide_tvar_baz ⇒ (_) hide_tvar_baz ⇒ (_) hide_tvar_baz"
where "hide_tvar_g a b = a"
text‹Instead of specifying the type variables explicitely. This makes, in particular
for type constructors with a large number of type variables, definitions much
more concise. This syntax is also used in the output of antiquotations, e.g.,
@{term[show_types] "x = hide_tvar_g"}. Both the print translation and the parse
translation can be disabled for each type individually:›
update_default_tvars_mode "_ hide_tvar_foobar" (noprint,noparse)
update_default_tvars_mode "_ hide_tvar_foobar" (noprint,noparse)
text‹ Now, Isabelle's interactive output and the antiquotations will show
all type variables, e.g., @{term[show_types] "x = hide_tvar_g"}.›
end
Theory Ref
section‹References›
text‹
This theory, we introduce a generic reference. All our typed pointers include such
a reference, which allows us to distinguish pointers of the same type, but also to
iterate over all pointers in a set.›
theory
Ref
imports
"HOL-Library.Adhoc_Overloading"
"../preliminaries/Hiding_Type_Variables"
begin
instantiation sum :: (linorder, linorder) linorder
begin
definition less_eq_sum :: "'a + 'b ⇒ 'a + 'b ⇒ bool"
where
"less_eq_sum t t' = (case t of
Inl l ⇒ (case t' of
Inl l' ⇒ l ≤ l'
| Inr r' ⇒ True)
| Inr r ⇒ (case t' of
Inl l' ⇒ False
| Inr r' ⇒ r ≤ r'))"
definition less_sum :: "'a + 'b ⇒ 'a + 'b ⇒ bool"
where
"less_sum t t' ≡ t ≤ t' ∧ ¬ t' ≤ t"
instance by(standard) (auto simp add: less_eq_sum_def less_sum_def split: sum.splits)
end
type_synonym ref = nat
consts cast :: 'a
end
Theory Core_DOM_Basic_Datatypes
section‹Basic Data Types›
text‹
\label{sec:Core_DOM_Basic_Datatypes}
This theory formalizes the primitive data types used by the DOM standard~\cite{dom-specification}.
›
theory Core_DOM_Basic_Datatypes
imports
Main
begin
type_synonym USVString = string
text‹
In the official standard, the type @{type "USVString"} corresponds to the set of all possible
sequences of Unicode scalar values. As we are not interested in analyzing the specifics of Unicode
strings, we just model @{type "USVString"} using the standard type @{type "string"} of Isabelle/HOL.
›
type_synonym DOMString = string
text‹
In the official standard, the type @{type "DOMString"} corresponds to the set of all possible
sequences of code units, commonly interpreted as UTF-16 encoded strings. Again, as we are not
interested in analyzing the specifics of Unicode strings, we just model @{type "DOMString"} using
the standard type @{type "string"} of Isabelle/HOL.
›
type_synonym doctype = DOMString
paragraph‹Examples›
definition html :: doctype
where "html = ''<!DOCTYPE html>''"
hide_const id
text ‹This dummy locale is used to create scoped definitions by using global interpretations
and defines.›
locale l_dummy
end
Theory BaseClass
section‹The Class Infrastructure›
text‹In this theory, we introduce the basic infrastructure for our encoding
of classes.›
theory BaseClass
imports
"HOL-Library.Finite_Map"
"../pointers/Ref"
"../Core_DOM_Basic_Datatypes"
begin
named_theorems instances
consts get :: 'a
consts put :: 'a
consts delete :: 'a
text ‹Overall, the definition of the class types follows closely the one of the pointer
types. Instead of datatypes, we use records for our classes. This allows us to, first,
make use of record inheritance, which is, in addition to the type synonyms of
previous class types, the second place where the inheritance relationship of
our types manifest. Second, we get a convenient notation to define classes, in
addition to automatically generated getter and setter functions.›
text ‹Along with our class types, we also develop our heap type, which is a finite
map at its core. It is important to note that while the map stores a mapping
from @{term "object_ptr"} to @{term "Object"}, we restrict the type variables
of the record extension slot of @{term "Object"} in such a way that allows
down-casting, but requires a bit of taking-apart and re-assembling of our records
before they are stored in the heap.›
text ‹Throughout the theory files, we will use underscore case to reference pointer
types, and camel case for class types.›
text ‹Every class type contains at least one attribute; nothing. This is used for
two purposes: first, the record package does not allow records without any
attributes. Second, we will use the getter of nothing later to check whether a
class of the correct type could be retrieved, for which we will be able to use
our infrastructure regarding the behaviour of getters across different heaps.›
locale l_type_wf = fixes type_wf :: "'heap ⇒ bool"
locale l_known_ptr = fixes known_ptr :: "'ptr ⇒ bool"
end
Theory Heap_Error_Monad
section‹The Heap Error Monad›
text ‹In this theory, we define a heap and error monad for modeling exceptions.
This allows us to define composite methods similar to stateful programming in Haskell,
but also to stay close to the official DOM specification.›
theory
Heap_Error_Monad
imports
Hiding_Type_Variables
"HOL-Library.Monad_Syntax"
begin
subsection ‹The Program Data Type›
datatype ('heap, 'e, 'result) prog = Prog (the_prog: "'heap ⇒ 'e + 'result × 'heap")
register_default_tvars "('heap, 'e, 'result) prog" (print, parse)
subsection ‹Basic Functions›
definition
bind :: "(_, 'result) prog ⇒ ('result ⇒ (_, 'result2) prog) ⇒ (_, 'result2) prog"
where
"bind f g = Prog (λh. (case (the_prog f) h of Inr (x, h') ⇒ (the_prog (g x)) h'
| Inl exception ⇒ Inl exception))"
adhoc_overloading Monad_Syntax.bind bind
definition
execute :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ ('e + 'result × 'heap)"
("((_)/ ⊢ (_))" [51, 52] 55)
where
"execute h p = (the_prog p) h"
definition
returns_result :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ 'result ⇒ bool"
("((_)/ ⊢ (_)/ →⇩r (_))" [60, 35, 61] 65)
where
"returns_result h p r ⟷ (case h ⊢ p of Inr (r', _) ⇒ r = r' | Inl _ ⇒ False)"
fun select_result ("|(_)|⇩r")
where
"select_result (Inr (r, _)) = r"
| "select_result (Inl _) = undefined"
lemma returns_result_eq [elim]: "h ⊢ f →⇩r y ⟹ h ⊢ f →⇩r y' ⟹ y = y'"
by(auto simp add: returns_result_def split: sum.splits)
definition
returns_heap :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ 'heap ⇒ bool"
("((_)/ ⊢ (_)/ →⇩h (_))" [60, 35, 61] 65)
where
"returns_heap h p h' ⟷ (case h ⊢ p of Inr (_ , h'') ⇒ h' = h'' | Inl _ ⇒ False)"
fun select_heap ("|(_)|⇩h")
where
"select_heap (Inr ( _, h)) = h"
| "select_heap (Inl _) = undefined"
lemma returns_heap_eq [elim]: "h ⊢ f →⇩h h' ⟹ h ⊢ f →⇩h h'' ⟹ h' = h''"
by(auto simp add: returns_heap_def split: sum.splits)
definition
returns_result_heap :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ 'result ⇒ 'heap ⇒ bool"
("((_)/ ⊢ (_)/ →⇩r (_) →⇩h (_))" [60, 35, 61, 62] 65)
where
"returns_result_heap h p r h' ⟷ h ⊢ p →⇩r r ∧ h ⊢ p →⇩h h'"
lemma return_result_heap_code [code]:
"returns_result_heap h p r h' ⟷ (case h ⊢ p of Inr (r', h'') ⇒ r = r' ∧ h' = h'' | Inl _ ⇒ False)"
by(auto simp add: returns_result_heap_def returns_result_def returns_heap_def split: sum.splits)
fun select_result_heap ("|(_)|⇩r⇩h")
where
"select_result_heap (Inr (r, h)) = (r, h)"
| "select_result_heap (Inl _) = undefined"
definition
returns_error :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ 'e ⇒ bool"
("((_)/ ⊢ (_)/ →⇩e (_))" [60, 35, 61] 65)
where
"returns_error h p e = (case h ⊢ p of Inr _ ⇒ False | Inl e' ⇒ e = e')"
definition is_OK :: "'heap ⇒ ('heap, 'e, 'result) prog ⇒ bool" ("((_)/ ⊢ ok (_))" [75, 75])
where
"is_OK h p = (case h ⊢ p of Inr _ ⇒ True | Inl _ ⇒ False)"
lemma is_OK_returns_result_I [intro]: "h ⊢ f →⇩r y ⟹ h ⊢ ok f"
by(auto simp add: is_OK_def returns_result_def split: sum.splits)
lemma is_OK_returns_result_E [elim]:
assumes "h ⊢ ok f"
obtains x where "h ⊢ f →⇩r x"
using assms by(auto simp add: is_OK_def returns_result_def split: sum.splits)
lemma is_OK_returns_heap_I [intro]: "h ⊢ f →⇩h h' ⟹ h ⊢ ok f"
by(auto simp add: is_OK_def returns_heap_def split: sum.splits)
lemma is_OK_returns_heap_E [elim]:
assumes "h ⊢ ok f"
obtains h' where "h ⊢ f →⇩h h'"
using assms by(auto simp add: is_OK_def returns_heap_def split: sum.splits)
lemma select_result_I:
assumes "h ⊢ ok f"
and "⋀x. h ⊢ f →⇩r x ⟹ P x"
shows "P |h ⊢ f|⇩r"
using assms
by(auto simp add: is_OK_def returns_result_def split: sum.splits)
lemma select_result_I2 [simp]:
assumes "h ⊢ f →⇩r x"
shows "|h ⊢ f|⇩r = x"
using assms
by(auto simp add: is_OK_def returns_result_def split: sum.splits)
lemma returns_result_select_result [simp]:
assumes "h ⊢ ok f"
shows "h ⊢ f →⇩r |h ⊢ f|⇩r"
using assms
by (simp add: select_result_I)
lemma select_result_E:
assumes "P |h ⊢ f|⇩r" and "h ⊢ ok f"
obtains x where "h ⊢ f →⇩r x" and "P x"
using assms
by(auto simp add: is_OK_def returns_result_def split: sum.splits)
lemma select_result_eq: "(⋀x .h ⊢ f →⇩r x = h' ⊢ f →⇩r x) ⟹ |h ⊢ f|⇩r = |h' ⊢ f|⇩r"
by (metis (no_types, lifting) is_OK_def old.sum.simps(6) select_result.elims
select_result_I select_result_I2)
definition error :: "'e ⇒ ('heap, 'e, 'result) prog"
where
"error exception = Prog (λh. Inl exception)"
lemma error_bind [iff]: "(error e ⤜ g) = error e"
unfolding error_def bind_def by auto
lemma error_returns_result [simp]: "¬ (h ⊢ error e →⇩r y)"
unfolding returns_result_def error_def execute_def by auto
lemma error_returns_heap [simp]: "¬ (h ⊢ error e →⇩h h')"
unfolding returns_heap_def error_def execute_def by auto
lemma error_returns_error [simp]: "h ⊢ error e →⇩e e"
unfolding returns_error_def error_def execute_def by auto
definition return :: "'result ⇒ ('heap, 'e, 'result) prog"
where
"return result = Prog (λh. Inr (result, h))"
lemma return_ok [simp]: "h ⊢ ok (return x)"
by(simp add: return_def is_OK_def execute_def)
lemma return_bind [iff]: "(return x ⤜ g) = g x"
unfolding return_def bind_def by auto
lemma return_id [simp]: "f ⤜ return = f"
by (induct f) (auto simp add: return_def bind_def split: sum.splits prod.splits)
lemma return_returns_result [iff]: "(h ⊢ return x →⇩r y) = (x = y)"
unfolding returns_result_def return_def execute_def by auto
lemma return_returns_heap [iff]: "(h ⊢ return x →⇩h h') = (h = h')"
unfolding returns_heap_def return_def execute_def by auto
lemma return_returns_error [iff]: "¬ h ⊢ return x →⇩e e"
unfolding returns_error_def execute_def return_def by auto
definition noop :: "('heap, 'e, unit) prog"
where
"noop = return ()"
lemma noop_returns_heap [simp]: "h ⊢ noop →⇩h h' ⟷ h = h'"
by(simp add: noop_def)
definition get_heap :: "('heap, 'e, 'heap) prog"
where
"get_heap = Prog (λh. h ⊢ return h)"
lemma get_heap_ok [simp]: "h ⊢ ok (get_heap)"
by (simp add: get_heap_def execute_def is_OK_def return_def)
lemma get_heap_returns_result [simp]: "(h ⊢ get_heap ⤜ (λh'. f h') →⇩r x) = (h ⊢ f h →⇩r x)"
by(simp add: get_heap_def returns_result_def bind_def return_def execute_def)
lemma get_heap_returns_heap [simp]: "(h ⊢ get_heap ⤜ (λh'. f h') →⇩h h'') = (h ⊢ f h →⇩h h'')"
by(simp add: get_heap_def returns_heap_def bind_def return_def execute_def)
lemma get_heap_is_OK [simp]: "(h ⊢ ok (get_heap ⤜ (λh'. f h'))) = (h ⊢ ok (f h))"
by(auto simp add: get_heap_def is_OK_def bind_def return_def execute_def)
lemma get_heap_E [elim]: "(h ⊢ get_heap →⇩r x) ⟹ x = h"
by(simp add: get_heap_def returns_result_def return_def execute_def)
definition return_heap :: "'heap ⇒ ('heap, 'e, unit) prog"
where
"return_heap h = Prog (λ_. h ⊢ return ())"
lemma return_heap_E [iff]: "(h ⊢ return_heap h' →⇩h h'') = (h'' = h')"
by(simp add: return_heap_def returns_heap_def return_def execute_def)
lemma return_heap_returns_result [simp]: "h ⊢ return_heap h' →⇩r ()"
by(simp add: return_heap_def execute_def returns_result_def return_def)
subsection ‹Pure Heaps›
definition pure :: "('heap, 'e, 'result) prog ⇒ 'heap ⇒ bool"
where "pure f h ⟷ h ⊢ ok f ⟶ h ⊢ f →⇩h h"
lemma return_pure [simp]: "pure (return x) h"
by(simp add: pure_def return_def is_OK_def returns_heap_def execute_def)
lemma error_pure [simp]: "pure (error e) h"
by(simp add: pure_def error_def is_OK_def returns_heap_def execute_def)
lemma noop_pure [simp]: "pure (noop) h"
by (simp add: noop_def)
lemma get_pure [simp]: "pure get_heap h"
by(simp add: pure_def get_heap_def is_OK_def returns_heap_def return_def execute_def)
lemma pure_returns_heap_eq:
"h ⊢ f →⇩h h' ⟹ pure f h ⟹ h = h'"
by (meson pure_def is_OK_returns_heap_I returns_heap_eq)
lemma pure_eq_iff:
"(∀h' x. h ⊢ f →⇩r x ⟶ h ⊢ f →⇩h h' ⟶ h = h') ⟷ pure f h"
by(auto simp add: pure_def)
subsection ‹Bind›
lemma bind_assoc [simp]:
"((bind f g) ⤜ h) = (f ⤜ (λx. (g x ⤜ h)))"
by(auto simp add: bind_def split: sum.splits)
lemma bind_returns_result_E:
assumes "h ⊢ f ⤜ g →⇩r y"
obtains x h' where "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩r y"
using assms by(auto simp add: bind_def returns_result_def returns_heap_def execute_def
split: sum.splits)
lemma bind_returns_result_E2:
assumes "h ⊢ f ⤜ g →⇩r y" and "pure f h"
obtains x where "h ⊢ f →⇩r x" and "h ⊢ g x →⇩r y"
using assms pure_returns_heap_eq bind_returns_result_E by metis
lemma bind_returns_result_E3:
assumes "h ⊢ f ⤜ g →⇩r y" and "h ⊢ f →⇩r x" and "pure f h"
shows "h ⊢ g x →⇩r y"
using assms returns_result_eq bind_returns_result_E2 by metis
lemma bind_returns_result_E4:
assumes "h ⊢ f ⤜ g →⇩r y" and "h ⊢ f →⇩r x"
obtains h' where "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩r y"
using assms returns_result_eq bind_returns_result_E by metis
lemma bind_returns_heap_E:
assumes "h ⊢ f ⤜ g →⇩h h''"
obtains x h' where "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩h h''"
using assms by(auto simp add: bind_def returns_result_def returns_heap_def execute_def
split: sum.splits)
lemma bind_returns_heap_E2 [elim]:
assumes "h ⊢ f ⤜ g →⇩h h'" and "pure f h"
obtains x where "h ⊢ f →⇩r x" and "h ⊢ g x →⇩h h'"
using assms pure_returns_heap_eq by (fastforce elim: bind_returns_heap_E)
lemma bind_returns_heap_E3 [elim]:
assumes "h ⊢ f ⤜ g →⇩h h'" and "h ⊢ f →⇩r x" and "pure f h"
shows "h ⊢ g x →⇩h h'"
using assms pure_returns_heap_eq returns_result_eq by (fastforce elim: bind_returns_heap_E)
lemma bind_returns_heap_E4:
assumes "h ⊢ f ⤜ g →⇩h h''" and "h ⊢ f →⇩h h'"
obtains x where "h ⊢ f →⇩r x" and "h' ⊢ g x →⇩h h''"
using assms
by (metis bind_returns_heap_E returns_heap_eq)
lemma bind_returns_error_I [intro]:
assumes "h ⊢ f →⇩e e"
shows "h ⊢ f ⤜ g →⇩e e"
using assms
by(auto simp add: returns_error_def bind_def execute_def split: sum.splits)
lemma bind_returns_error_I3:
assumes "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩e e"
shows "h ⊢ f ⤜ g →⇩e e"
using assms
by(auto simp add: returns_error_def bind_def execute_def returns_heap_def returns_result_def
split: sum.splits)
lemma bind_returns_error_I2 [intro]:
assumes "pure f h" and "h ⊢ f →⇩r x" and "h ⊢ g x →⇩e e"
shows "h ⊢ f ⤜ g →⇩e e"
using assms
by (meson bind_returns_error_I3 is_OK_returns_result_I pure_def)
lemma bind_is_OK_E [elim]:
assumes "h ⊢ ok (f ⤜ g)"
obtains x h' where "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ ok (g x)"
using assms
by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def
split: sum.splits)
lemma bind_is_OK_E2:
assumes "h ⊢ ok (f ⤜ g)" and "h ⊢ f →⇩r x"
obtains h' where "h ⊢ f →⇩h h'" and "h' ⊢ ok (g x)"
using assms
by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def
split: sum.splits)
lemma bind_returns_result_I [intro]:
assumes "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩r y"
shows "h ⊢ f ⤜ g →⇩r y"
using assms
by(auto simp add: bind_def returns_result_def returns_heap_def execute_def
split: sum.splits)
lemma bind_pure_returns_result_I [intro]:
assumes "pure f h" and "h ⊢ f →⇩r x" and "h ⊢ g x →⇩r y"
shows "h ⊢ f ⤜ g →⇩r y"
using assms
by (meson bind_returns_result_I pure_def is_OK_returns_result_I)
lemma bind_pure_returns_result_I2 [intro]:
assumes "pure f h" and "h ⊢ ok f" and "⋀x. h ⊢ f →⇩r x ⟹ h ⊢ g x →⇩r y"
shows "h ⊢ f ⤜ g →⇩r y"
using assms by auto
lemma bind_returns_heap_I [intro]:
assumes "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ g x →⇩h h''"
shows "h ⊢ f ⤜ g →⇩h h''"
using assms
by(auto simp add: bind_def returns_result_def returns_heap_def execute_def
split: sum.splits)
lemma bind_returns_heap_I2 [intro]:
assumes "h ⊢ f →⇩h h'" and "⋀x. h ⊢ f →⇩r x ⟹ h' ⊢ g x →⇩h h''"
shows "h ⊢ f ⤜ g →⇩h h''"
using assms
by (meson bind_returns_heap_I is_OK_returns_heap_I is_OK_returns_result_E)
lemma bind_is_OK_I [intro]:
assumes "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'" and "h' ⊢ ok (g x)"
shows "h ⊢ ok (f ⤜ g)"
by (meson assms(1) assms(2) assms(3) bind_returns_heap_I is_OK_returns_heap_E
is_OK_returns_heap_I)
lemma bind_is_OK_I2 [intro]:
assumes "h ⊢ ok f" and "⋀x h'. h ⊢ f →⇩r x ⟹ h ⊢ f →⇩h h' ⟹ h' ⊢ ok (g x)"
shows "h ⊢ ok (f ⤜ g)"
using assms by blast
lemma bind_is_OK_pure_I [intro]:
assumes "pure f h" and "h ⊢ ok f" and "⋀x. h ⊢ f →⇩r x ⟹ h ⊢ ok (g x)"
shows "h ⊢ ok (f ⤜ g)"
using assms by blast
lemma bind_pure_I:
assumes "pure f h" and "⋀x. h ⊢ f →⇩r x ⟹ pure (g x) h"
shows "pure (f ⤜ g) h"
using assms
by (metis bind_returns_heap_E2 pure_def pure_returns_heap_eq is_OK_returns_heap_E)
lemma pure_pure:
assumes "h ⊢ ok f" and "pure f h"
shows "h ⊢ f →⇩h h"
using assms returns_heap_eq
unfolding pure_def
by auto
lemma bind_returns_error_eq:
assumes "h ⊢ f →⇩e e"
and "h ⊢ g →⇩e e"
shows "h ⊢ f = h ⊢ g"
using assms
by(auto simp add: returns_error_def split: sum.splits)
subsection ‹Map›
fun map_M :: "('x ⇒ ('heap, 'e, 'result) prog) ⇒ 'x list ⇒ ('heap, 'e, 'result list) prog"
where
"map_M f [] = return []"
| "map_M f (x#xs) = do {
y ← f x;
ys ← map_M f xs;
return (y # ys)
}"
lemma map_M_ok_I [intro]:
"(⋀x. x ∈ set xs ⟹ h ⊢ ok (f x)) ⟹ (⋀x. x ∈ set xs ⟹ pure (f x) h) ⟹ h ⊢ ok (map_M f xs)"
apply(induct xs)
by (simp_all add: bind_is_OK_I2 bind_is_OK_pure_I)
lemma map_M_pure_I : "⋀h. (⋀x. x ∈ set xs ⟹ pure (f x) h) ⟹ pure (map_M f xs) h"
apply(induct xs)
apply(simp)
by(auto intro!: bind_pure_I)
lemma map_M_pure_E :
assumes "h ⊢ map_M g xs →⇩r ys" and "x ∈ set xs" and "⋀x h. x ∈ set xs ⟹ pure (g x) h"
obtains y where "h ⊢ g x →⇩r y" and "y ∈ set ys"
apply(insert assms, induct xs arbitrary: ys)
apply(simp)
apply(auto elim!: bind_returns_result_E)[1]
by (metis (full_types) pure_returns_heap_eq)
lemma map_M_pure_E2:
assumes "h ⊢ map_M g xs →⇩r ys" and "y ∈ set ys" and "⋀x h. x ∈ set xs ⟹ pure (g x) h"
obtains x where "h ⊢ g x →⇩r y" and "x ∈ set xs"
apply(insert assms, induct xs arbitrary: ys)
apply(simp)
apply(auto elim!: bind_returns_result_E)[1]
by (metis (full_types) pure_returns_heap_eq)
subsection ‹Forall›
fun forall_M :: "('y ⇒ ('heap, 'e, 'result) prog) ⇒ 'y list ⇒ ('heap, 'e, unit) prog"
where
"forall_M P [] = return ()"
| "forall_M P (x # xs) = do {
P x;
forall_M P xs
}"
lemma pure_forall_M_I: "(⋀x. x ∈ set xs ⟹ pure (P x) h) ⟹ pure (forall_M P xs) h"
apply(induct xs)
by(auto intro!: bind_pure_I)
subsection ‹Fold›
fun fold_M :: "('result ⇒ 'y ⇒ ('heap, 'e, 'result) prog) ⇒ 'result ⇒ 'y list
⇒ ('heap, 'e, 'result) prog"
where
"fold_M f d [] = return d" |
"fold_M f d (x # xs) = do { y ← f d x; fold_M f y xs }"
lemma fold_M_pure_I : "(⋀d x. pure (f d x) h) ⟹ (⋀d. pure (fold_M f d xs) h)"
apply(induct xs)
by(auto intro: bind_pure_I)
subsection ‹Filter›
fun filter_M :: "('x ⇒ ('heap, 'e, bool) prog) ⇒ 'x list ⇒ ('heap, 'e, 'x list) prog"
where
"filter_M P [] = return []"
| "filter_M P (x#xs) = do {
p ← P x;
ys ← filter_M P xs;
return (if p then x # ys else ys)
}"
lemma filter_M_pure_I [intro]: "(⋀x. x ∈ set xs ⟹ pure (P x) h) ⟹ pure (filter_M P xs)h"
apply(induct xs)
by(auto intro!: bind_pure_I)
lemma filter_M_is_OK_I [intro]:
"(⋀x. x ∈ set xs ⟹ h ⊢ ok (P x)) ⟹ (⋀x. x ∈ set xs ⟹ pure (P x) h) ⟹ h ⊢ ok (filter_M P xs)"
apply(induct xs)
apply(simp)
by(auto intro!: bind_is_OK_pure_I)
lemma filter_M_not_more_elements:
assumes "h ⊢ filter_M P xs →⇩r ys" and "⋀x. x ∈ set xs ⟹ pure (P x) h" and "x ∈ set ys"
shows "x ∈ set xs"
apply(insert assms, induct xs arbitrary: ys)
by(auto elim!: bind_returns_result_E2 split: if_splits intro!: set_ConsD)
lemma filter_M_in_result_if_ok:
assumes "h ⊢ filter_M P xs →⇩r ys" and "⋀h x. x ∈ set xs ⟹ pure (P x) h" and "x ∈ set xs" and
"h ⊢ P x →⇩r True"
shows "x ∈ set ys"
apply(insert assms, induct xs arbitrary: ys)
apply(simp)
apply(auto elim!: bind_returns_result_E2)[1]
by (metis returns_result_eq)
lemma filter_M_holds_for_result:
assumes "h ⊢ filter_M P xs →⇩r ys" and "x ∈ set ys" and "⋀x h. x ∈ set xs ⟹ pure (P x) h"
shows "h ⊢ P x →⇩r True"
apply(insert assms, induct xs arbitrary: ys)
by(auto elim!: bind_returns_result_E2 split: if_splits intro!: set_ConsD)
lemma filter_M_empty_I:
assumes "⋀x. pure (P x) h"
and "∀x ∈ set xs. h ⊢ P x →⇩r False"
shows "h ⊢ filter_M P xs →⇩r []"
using assms
apply(induct xs)
by(auto intro!: bind_pure_returns_result_I)
lemma filter_M_subset_2: "h ⊢ filter_M P xs →⇩r ys ⟹ h' ⊢ filter_M P xs →⇩r ys'
⟹ (⋀x. pure (P x) h) ⟹ (⋀x. pure (P x) h')
⟹ (∀b. ∀x ∈ set xs. h ⊢ P x →⇩r True ⟶ h' ⊢ P x →⇩r b ⟶ b)
⟹ set ys ⊆ set ys'"
proof -
assume 1: "h ⊢ filter_M P xs →⇩r ys" and 2: "h' ⊢ filter_M P xs →⇩r ys'"
and 3: "(⋀x. pure (P x) h)" and "(⋀x. pure (P x) h')"
and 4: "∀b. ∀x∈set xs. h ⊢ P x →⇩r True ⟶ h' ⊢ P x →⇩r b ⟶ b"
have h1: "∀x ∈ set xs. h' ⊢ ok (P x)"
using 2 3 ‹(⋀x. pure (P x) h')›
apply(induct xs arbitrary: ys')
by(auto elim!: bind_returns_result_E2)
then have 5: "∀x∈set xs. h ⊢ P x →⇩r True ⟶ h' ⊢ P x →⇩r True"
using 4
apply(auto)[1]
by (metis is_OK_returns_result_E)
show ?thesis
using 1 2 3 5 ‹(⋀x. pure (P x) h')›
apply(induct xs arbitrary: ys ys')
apply(auto)[1]
apply(auto elim!: bind_returns_result_E2 split: if_splits)[1]
apply auto[1]
apply auto[1]
apply(metis returns_result_eq)
apply auto[1]
apply auto[1]
apply auto[1]
by(auto)
qed
lemma filter_M_subset: "h ⊢ filter_M P xs →⇩r ys ⟹ set ys ⊆ set xs"
apply(induct xs arbitrary: h ys)
apply(auto)[1]
apply(auto elim!: bind_returns_result_E split: if_splits)[1]
apply blast
by blast
lemma filter_M_distinct: "h ⊢ filter_M P xs →⇩r ys ⟹ distinct xs ⟹ distinct ys"
apply(induct xs arbitrary: h ys)
apply(auto)[1]
using filter_M_subset
apply(auto elim!: bind_returns_result_E)[1]
by fastforce
lemma filter_M_filter: "h ⊢ filter_M P xs →⇩r ys ⟹ (⋀x. x ∈ set xs ⟹ pure (P x) h)
⟹ (∀x ∈ set xs. h ⊢ ok P x) ∧ ys = filter (λx. |h ⊢ P x|⇩r) xs"
apply(induct xs arbitrary: ys)
by(auto elim!: bind_returns_result_E2)
lemma filter_M_filter2: "(⋀x. x ∈ set xs ⟹ pure (P x) h ∧ h ⊢ ok P x)
⟹ filter (λx. |h ⊢ P x|⇩r) xs = ys ⟹ h ⊢ filter_M P xs →⇩r ys"
apply(induct xs arbitrary: ys)
by(auto elim!: bind_returns_result_E2 intro!: bind_pure_returns_result_I)
lemma filter_ex1: "∃!x ∈ set xs. P x ⟹ P x ⟹ x ∈ set xs ⟹ distinct xs
⟹ filter P xs = [x]"
apply(auto)[1]
apply(induct xs)
apply(auto)[1]
apply(auto)[1]
using filter_empty_conv by fastforce
lemma filter_M_ex1:
assumes "h ⊢ filter_M P xs →⇩r ys"
and "x ∈ set xs"
and "∃!x ∈ set xs. h ⊢ P x →⇩r True"
and "⋀x. x ∈ set xs ⟹ pure (P x) h"
and "distinct xs"
and "h ⊢ P x →⇩r True"
shows "ys = [x]"
proof -
have *: "∃!x ∈ set xs. |h ⊢ P x|⇩r"
apply(insert assms(1) assms(3) assms(4))
apply(drule filter_M_filter)
apply(simp)
apply(auto simp add: select_result_I2)[1]
by (metis (full_types) is_OK_returns_result_E select_result_I2)
then show ?thesis
apply(insert assms(1) assms(4))
apply(drule filter_M_filter)
apply(auto)[1]
by (metis * assms(2) assms(5) assms(6) distinct_filter
distinct_length_2_or_more filter_empty_conv filter_set list.exhaust
list.set_intros(1) list.set_intros(2) member_filter select_result_I2)
qed
lemma filter_M_eq:
assumes "⋀x. pure (P x) h" and "⋀x. pure (P x) h'"
and "⋀b x. x ∈ set xs ⟹ h ⊢ P x →⇩r b = h' ⊢ P x →⇩r b"
shows "h ⊢ filter_M P xs →⇩r ys ⟷ h' ⊢ filter_M P xs →⇩r ys"
using assms
apply (induct xs arbitrary: ys)
by(auto elim!: bind_returns_result_E2 intro!: bind_pure_returns_result_I
dest: returns_result_eq)
subsection ‹Map Filter›
definition map_filter_M :: "('x ⇒ ('heap, 'e, 'y option) prog) ⇒ 'x list
⇒ ('heap, 'e, 'y list) prog"
where
"map_filter_M f xs = do {
ys_opts ← map_M f xs;
ys_no_opts ← filter_M (λx. return (x ≠ None)) ys_opts;
map_M (λx. return (the x)) ys_no_opts
}"
lemma map_filter_M_pure: "(⋀x h. x ∈ set xs ⟹ pure (f x) h) ⟹ pure (map_filter_M f xs) h"
by(auto simp add: map_filter_M_def map_M_pure_I intro!: bind_pure_I)
lemma map_filter_M_pure_E:
assumes "h ⊢ (map_filter_M::('x ⇒ ('heap, 'e, 'y option) prog) ⇒ 'x list
⇒ ('heap, 'e, 'y list) prog) f xs →⇩r ys" and "y ∈ set ys" and "⋀x h. x ∈ set xs ⟹ pure (f x) h"
obtains x where "h ⊢ f x →⇩r Some y" and "x ∈ set xs"
proof -
obtain ys_opts ys_no_opts where
ys_opts: "h ⊢ map_M f xs →⇩r ys_opts" and
ys_no_opts: "h ⊢ filter_M (λx. (return (x ≠ None)::('heap, 'e, bool) prog)) ys_opts →⇩r ys_no_opts" and
ys: "h ⊢ map_M (λx. (return (the x)::('heap, 'e, 'y) prog)) ys_no_opts →⇩r ys"
using assms
by(auto simp add: map_filter_M_def map_M_pure_I elim!: bind_returns_result_E2)
have "∀y ∈ set ys_no_opts. y ≠ None"
using ys_no_opts filter_M_holds_for_result
by fastforce
then have "Some y ∈ set ys_no_opts"
using map_M_pure_E2 ys ‹y ∈ set ys›
by (metis (no_types, lifting) option.collapse return_pure return_returns_result)
then have "Some y ∈ set ys_opts"
using filter_M_subset ys_no_opts by fastforce
then show "(⋀x. h ⊢ f x →⇩r Some y ⟹ x ∈ set xs ⟹ thesis) ⟹ thesis"
by (metis assms(3) map_M_pure_E2 ys_opts)
qed
subsection ‹Iterate›
fun iterate_M :: "('heap, 'e, 'result) prog list ⇒ ('heap, 'e, 'result) prog"
where
"iterate_M [] = return undefined"
| "iterate_M (x # xs) = x ⤜ (λ_. iterate_M xs)"
lemma iterate_M_concat:
assumes "h ⊢ iterate_M xs →⇩h h'"
and "h' ⊢ iterate_M ys →⇩h h''"
shows "h ⊢ iterate_M (xs @ ys) →⇩h h''"
using assms
apply(induct "xs" arbitrary: h h'')
apply(simp)
apply(auto)[1]
by (meson bind_returns_heap_E bind_returns_heap_I)
subsection‹Miscellaneous Rules›
lemma execute_bind_simp:
assumes "h ⊢ f →⇩r x" and "h ⊢ f →⇩h h'"
shows "h ⊢ f ⤜ g = h' ⊢ g x"
using assms
by(auto simp add: returns_result_def returns_heap_def bind_def execute_def
split: sum.splits)
lemma bind_cong [fundef_cong]:
fixes f1 f2 :: "('heap, 'e, 'result) prog"
and g1 g2 :: "'result ⇒ ('heap, 'e, 'result2) prog"
assumes "h ⊢ f1 = h ⊢ f2"
and "⋀y h'. h ⊢ f1 →⇩r y ⟹ h ⊢ f1 →⇩h h' ⟹ h' ⊢ g1 y = h' ⊢ g2 y"
shows "h ⊢ (f1 ⤜ g1) = h ⊢ (f2 ⤜ g2)"
apply(insert assms, cases "h ⊢ f1")
by(auto simp add: bind_def returns_result_def returns_heap_def execute_def
split: sum.splits)
lemma bind_cong_2:
assumes "pure f h" and "pure f h'"
and "⋀x. h ⊢ f →⇩r x = h' ⊢ f →⇩r x"
and "⋀x. h ⊢ f →⇩r x ⟹ h ⊢ g x →⇩r y = h' ⊢ g x →⇩r y'"
shows "h ⊢ f ⤜ g →⇩r y = h' ⊢ f ⤜ g →⇩r y'"
using assms
by(auto intro!: bind_pure_returns_result_I elim!: bind_returns_result_E2)
lemma bind_case_cong [fundef_cong]:
assumes "x = x'" and "⋀a. x = Some a ⟹ f a h = f' a h"
shows "(case x of Some a ⇒ f a | None ⇒ g) h = (case x' of Some a ⇒ f' a | None ⇒ g) h"
by (insert assms, simp add: option.case_eq_if)
subsection ‹Reasoning About Reads and Writes›
definition preserved :: "('heap, 'e, 'result) prog ⇒ 'heap ⇒ 'heap ⇒ bool"
where
"preserved f h h' ⟷ (∀x. h ⊢ f →⇩r x ⟷ h' ⊢ f →⇩r x)"
lemma preserved_code [code]:
"preserved f h h' = (((h ⊢ ok f) ∧ (h' ⊢ ok f) ∧ |h ⊢ f|⇩r = |h' ⊢ f|⇩r) ∨ ((¬h ⊢ ok f) ∧ (¬h' ⊢ ok f)))"
apply(auto simp add: preserved_def)[1]
apply (meson is_OK_returns_result_E is_OK_returns_result_I)+
done
lemma reflp_preserved_f [simp]: "reflp (preserved f)"
by(auto simp add: preserved_def reflp_def)
lemma transp_preserved_f [simp]: "transp (preserved f)"
by(auto simp add: preserved_def transp_def)
definition
all_args :: "('a ⇒ ('heap, 'e, 'result) prog) ⇒ ('heap, 'e, 'result) prog set"
where
"all_args f = (⋃arg. {f arg})"
definition
reads :: "('heap ⇒ 'heap ⇒ bool) set ⇒ ('heap, 'e, 'result) prog ⇒ 'heap
⇒ 'heap ⇒ bool"
where
"reads S getter h h' ⟷ (∀P ∈ S. reflp P ∧ transp P) ∧ ((∀P ∈ S. P h h')
⟶ preserved getter h h')"
lemma reads_singleton [simp]: "reads {preserved f} f h h'"
by(auto simp add: reads_def)
lemma reads_bind_pure:
assumes "pure f h" and "pure f h'"
and "reads S f h h'"
and "⋀x. h ⊢ f →⇩r x ⟹ reads S (g x) h h'"
shows "reads S (f ⤜ g) h h'"
using assms
by(auto simp add: reads_def pure_pure preserved_def
intro!: bind_pure_returns_result_I is_OK_returns_result_I
dest: pure_returns_heap_eq
elim!: bind_returns_result_E)
lemma reads_insert_writes_set_left:
"∀P ∈ S. reflp P ∧ transp P ⟹ reads {getter} f h h' ⟹ reads (insert getter S) f h h'"
unfolding reads_def by simp
lemma reads_insert_writes_set_right:
"reflp getter ⟹ transp getter ⟹ reads S f h h' ⟹ reads (insert getter S) f h h'"
unfolding reads_def by blast
lemma reads_subset:
"reads S f h h' ⟹ ∀P ∈ S' - S. reflp P ∧ transp P ⟹ S ⊆ S' ⟹ reads S' f h h'"
by(auto simp add: reads_def)
lemma return_reads [simp]: "reads {} (return x) h h'"
by(simp add: reads_def preserved_def)
lemma error_reads [simp]: "reads {} (error e) h h'"
by(simp add: reads_def preserved_def)
lemma noop_reads [simp]: "reads {} noop h h'"
by(simp add: reads_def noop_def preserved_def)
lemma filter_M_reads:
assumes "⋀x. x ∈ set xs ⟹ pure (P x) h" and "⋀x. x ∈ set xs ⟹ pure (P x) h'"
and "⋀x. x ∈ set xs ⟹ reads S (P x) h h'"
and "∀P ∈ S. reflp P ∧ transp P"
shows "reads S (filter_M P xs) h h'"
using assms
apply(induct xs)
by(auto intro: reads_subset[OF return_reads] intro!: reads_bind_pure)
definition writes ::
"('heap, 'e, 'result) prog set ⇒ ('heap, 'e, 'result2) prog ⇒ 'heap ⇒ 'heap ⇒ bool"
where
"writes S setter h h'
⟷ (h ⊢ setter →⇩h h' ⟶ (∃progs. set progs ⊆ S ∧ h ⊢ iterate_M progs →⇩h h'))"
lemma writes_singleton [simp]: "writes (all_args f) (f a) h h'"
apply(auto simp add: writes_def all_args_def)[1]
apply(rule exI[where x="[f a]"])
by(auto)
lemma writes_singleton2 [simp]: "writes {f} f h h'"
apply(auto simp add: writes_def all_args_def)[1]
apply(rule exI[where x="[f]"])
by(auto)
lemma writes_union_left_I:
assumes "writes S f h h'"
shows "writes (S ∪ S') f h h'"
using assms
by(auto simp add: writes_def)
lemma writes_union_right_I:
assumes "writes S' f h h'"
shows "writes (S ∪ S') f h h'"
using assms
by(auto simp add: writes_def)
lemma writes_union_minus_split:
assumes "writes (S - S2) f h h'"
and "writes (S' - S2) f h h'"
shows "writes ((S ∪ S') - S2) f h h'"
using assms
by(auto simp add: writes_def)
lemma writes_subset: "writes S f h h' ⟹ S ⊆ S' ⟹ writes S' f h h'"
by(auto simp add: writes_def)
lemma writes_error [simp]: "writes S (error e) h h'"
by(simp add: writes_def)
lemma writes_not_ok [simp]: "¬h ⊢ ok f ⟹ writes S f h h'"
by(auto simp add: writes_def)
lemma writes_pure [simp]:
assumes "pure f h"
shows "writes S f h h'"
using assms
apply(auto simp add: writes_def)[1]
by (metis bot.extremum iterate_M.simps(1) list.set(1) pure_returns_heap_eq return_returns_heap)
lemma writes_bind:
assumes "⋀h2. writes S f h h2"
assumes "⋀x h2. h ⊢ f →⇩r x ⟹ h ⊢ f →⇩h h2 ⟹ writes S (g x) h2 h'"
shows "writes S (f ⤜ g) h h'"
using assms
apply(auto simp add: writes_def elim!: bind_returns_heap_E)[1]
by (metis iterate_M_concat le_supI set_append)
lemma writes_bind_pure:
assumes "pure f h"
assumes "⋀x. h ⊢ f →⇩r x ⟹ writes S (g x) h h'"
shows "writes S (f ⤜ g) h h'"
using assms
by(auto simp add: writes_def elim!: bind_returns_heap_E2)
lemma writes_small_big:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h' ⟹ P h h'"
assumes "reflp P"
assumes "transp P"
shows "P h h'"
proof -
obtain progs where "set progs ⊆ SW" and iterate: "h ⊢ iterate_M progs →⇩h h'"
by (meson assms(1) assms(2) writes_def)
then have "⋀h h'. ∀prog ∈ set progs. h ⊢ prog →⇩h h' ⟶ P h h'"
using assms(3) by auto
with iterate assms(4) assms(5) have "h ⊢ iterate_M progs →⇩h h' ⟹ P h h'"
proof(induct progs arbitrary: h)
case Nil
then show ?case
using reflpE by force
next
case (Cons a progs)
then show ?case
apply(auto elim!: bind_returns_heap_E)[1]
by (metis (full_types) transpD)
qed
then show ?thesis
using assms(1) iterate by blast
qed
lemma reads_writes_preserved:
assumes "reads SR getter h h'"
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h' ⟶ (∀r ∈ SR. r h h')"
shows "h ⊢ getter →⇩r x ⟷ h' ⊢ getter →⇩r x"
proof -
obtain progs where "set progs ⊆ SW" and iterate: "h ⊢ iterate_M progs →⇩h h'"
by (meson assms(2) assms(3) writes_def)
then have "⋀h h'. ∀prog ∈ set progs. h ⊢ prog →⇩h h' ⟶ (∀r ∈ SR. r h h')"
using assms(4) by blast
with iterate have "∀r ∈ SR. r h h'"
using writes_small_big assms(1) unfolding reads_def
by (metis assms(2) assms(3) assms(4))
then show ?thesis
using assms(1)
by (simp add: preserved_def reads_def)
qed
lemma reads_writes_separate_forwards:
assumes "reads SR getter h h'"
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "h ⊢ getter →⇩r x"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h' ⟶ (∀r ∈ SR. r h h')"
shows "h' ⊢ getter →⇩r x"
using reads_writes_preserved[OF assms(1) assms(2) assms(3) assms(5)] assms(4)
by(auto simp add: preserved_def)
lemma reads_writes_separate_backwards:
assumes "reads SR getter h h'"
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "h' ⊢ getter →⇩r x"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h' ⟶ (∀r ∈ SR. r h h')"
shows "h ⊢ getter →⇩r x"
using reads_writes_preserved[OF assms(1) assms(2) assms(3) assms(5)] assms(4)
by(auto simp add: preserved_def)
end
Theory BaseMonad
section‹The Monad Infrastructure›
text‹In this theory, we introduce the basic infrastructure for our monadic class encoding.›
theory BaseMonad
imports
"../classes/BaseClass"
"../preliminaries/Heap_Error_Monad"
begin
subsection‹Datatypes›
datatype exception = NotFoundError | HierarchyRequestError | NotSupportedError | SegmentationFault
| AssertException | NonTerminationException | InvokeError | TypeError
lemma finite_set_in [simp]: "x ∈ fset FS ⟷ x |∈| FS"
by (meson notin_fset)
consts put_M :: 'a
consts get_M :: 'a
consts delete_M :: 'a
lemma sorted_list_of_set_eq [dest]:
"sorted_list_of_set (fset x) = sorted_list_of_set (fset y) ⟹ x = y"
by (metis finite_fset fset_inject sorted_list_of_set(1))
locale l_ptr_kinds_M =
fixes ptr_kinds :: "'heap ⇒ 'ptr::linorder fset"
begin
definition a_ptr_kinds_M :: "('heap, exception, 'ptr list) prog"
where
"a_ptr_kinds_M = do {
h ← get_heap;
return (sorted_list_of_set (fset (ptr_kinds h)))
}"
lemma ptr_kinds_M_ok [simp]: "h ⊢ ok a_ptr_kinds_M"
by(simp add: a_ptr_kinds_M_def)
lemma ptr_kinds_M_pure [simp]: "pure a_ptr_kinds_M h"
by (auto simp add: a_ptr_kinds_M_def intro: bind_pure_I)
lemma ptr_kinds_ptr_kinds_M [simp]: "ptr ∈ set |h ⊢ a_ptr_kinds_M|⇩r ⟷ ptr |∈| ptr_kinds h"
by(simp add: a_ptr_kinds_M_def)
lemma ptr_kinds_M_ptr_kinds [simp]:
"h ⊢ a_ptr_kinds_M →⇩r xa ⟷ xa = sorted_list_of_set (fset (ptr_kinds h))"
by(auto simp add: a_ptr_kinds_M_def)
lemma ptr_kinds_M_ptr_kinds_returns_result [simp]:
"h ⊢ a_ptr_kinds_M ⤜ f →⇩r x ⟷ h ⊢ f (sorted_list_of_set (fset (ptr_kinds h))) →⇩r x"
by(auto simp add: a_ptr_kinds_M_def)
lemma ptr_kinds_M_ptr_kinds_returns_heap [simp]:
"h ⊢ a_ptr_kinds_M ⤜ f →⇩h h' ⟷ h ⊢ f (sorted_list_of_set (fset (ptr_kinds h))) →⇩h h'"
by(auto simp add: a_ptr_kinds_M_def)
end
locale l_get_M =
fixes get :: "'ptr ⇒ 'heap ⇒ 'obj option"
fixes type_wf :: "'heap ⇒ bool"
fixes ptr_kinds :: "'heap ⇒ 'ptr fset"
assumes "type_wf h ⟹ ptr |∈| ptr_kinds h ⟹ get ptr h ≠ None"
assumes "get ptr h ≠ None ⟹ ptr |∈| ptr_kinds h"
begin
definition a_get_M :: "'ptr ⇒ ('obj ⇒ 'result) ⇒ ('heap, exception, 'result) prog"
where
"a_get_M ptr getter = (do {
h ← get_heap;
(case get ptr h of
Some res ⇒ return (getter res)
| None ⇒ error SegmentationFault)
})"
lemma get_M_pure [simp]: "pure (a_get_M ptr getter) h"
by(auto simp add: a_get_M_def bind_pure_I split: option.splits)
lemma get_M_ok:
"type_wf h ⟹ ptr |∈| ptr_kinds h ⟹ h ⊢ ok (a_get_M ptr getter)"
apply(simp add: a_get_M_def)
by (metis l_get_M_axioms l_get_M_def option.case_eq_if return_ok)
lemma get_M_ptr_in_heap:
"h ⊢ ok (a_get_M ptr getter) ⟹ ptr |∈| ptr_kinds h"
apply(simp add: a_get_M_def)
by (metis error_returns_result is_OK_returns_result_E l_get_M_axioms l_get_M_def option.simps(4))
end
locale l_put_M = l_get_M get for get :: "'ptr ⇒ 'heap ⇒ 'obj option" +
fixes put :: "'ptr ⇒ 'obj ⇒ 'heap ⇒ 'heap"
begin
definition a_put_M :: "'ptr ⇒ (('v ⇒ 'v) ⇒ 'obj ⇒ 'obj) ⇒ 'v ⇒ ('heap, exception, unit) prog"
where
"a_put_M ptr setter v = (do {
obj ← a_get_M ptr id;
h ← get_heap;
return_heap (put ptr (setter (λ_. v) obj) h)
})"
lemma put_M_ok:
"type_wf h ⟹ ptr |∈| ptr_kinds h ⟹ h ⊢ ok (a_put_M ptr setter v)"
by(auto simp add: a_put_M_def intro!: bind_is_OK_I2 dest: get_M_ok elim!: bind_is_OK_E)
lemma put_M_ptr_in_heap:
"h ⊢ ok (a_put_M ptr setter v) ⟹ ptr |∈| ptr_kinds h"
by(auto simp add: a_put_M_def intro!: bind_is_OK_I2 elim: get_M_ptr_in_heap
dest: is_OK_returns_result_I elim!: bind_is_OK_E)
end
subsection ‹Setup for Defining Partial Functions›
lemma execute_admissible:
"ccpo.admissible (fun_lub (flat_lub (Inl (e::'e)))) (fun_ord (flat_ord (Inl e)))
((λa. ∀(h::'heap) h2 (r::'result). h ⊢ a = Inr (r, h2) ⟶ P h h2 r) ∘ Prog)"
proof (unfold comp_def, rule ccpo.admissibleI, clarify)
fix A :: "('heap ⇒ 'e + 'result × 'heap) set"
let ?lub = "Prog (fun_lub (flat_lub (Inl e)) A)"
fix h h2 r
assume 1: "Complete_Partial_Order.chain (fun_ord (flat_ord (Inl e))) A"
and 2: "∀xa∈A. ∀h h2 r. h ⊢ Prog xa = Inr (r, h2) ⟶ P h h2 r"
and 4: "h ⊢ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r, h2)"
have h1:"⋀a. Complete_Partial_Order.chain (flat_ord (Inl e)) {y. ∃f∈A. y = f a}"
by (rule chain_fun[OF 1])
show "P h h2 r"
using CollectD Inl_Inr_False prog.sel chain_fun[OF 1] flat_lub_in_chain[OF chain_fun[OF 1]] 2 4
unfolding execute_def fun_lub_def
proof -
assume a1: "the_prog (Prog (λx. flat_lub (Inl e) {y. ∃f∈A. y = f x})) h = Inr (r, h2)"
assume a2: "∀xa∈A. ∀h h2 r. the_prog (Prog xa) h = Inr (r, h2) ⟶ P h h2 r"
have "Inr (r, h2) ∈ {s. ∃f. f ∈ A ∧ s = f h} ∨ Inr (r, h2) = Inl e"
using a1 by (metis (lifting) ‹⋀aa a. flat_lub (Inl e) {y. ∃f∈A. y = f aa} = a ⟹ a = Inl e ∨ a ∈ {y. ∃f∈A. y = f aa}› prog.sel)
then show ?thesis
using a2 by fastforce
qed
qed
lemma execute_admissible2:
"ccpo.admissible (fun_lub (flat_lub (Inl (e::'e)))) (fun_ord (flat_ord (Inl e)))
((λa. ∀(h::'heap) h' h2 h2' (r::'result) r'.
h ⊢ a = Inr (r, h2) ⟶ h' ⊢ a = Inr (r', h2') ⟶ P h h' h2 h2' r r') ∘ Prog)"
proof (unfold comp_def, rule ccpo.admissibleI, clarify)
fix A :: "('heap ⇒ 'e + 'result × 'heap) set"
let ?lub = "Prog (fun_lub (flat_lub (Inl e)) A)"
fix h h' h2 h2' r r'
assume 1: "Complete_Partial_Order.chain (fun_ord (flat_ord (Inl e))) A"
and 2 [rule_format]: "∀xa∈A. ∀h h' h2 h2' r r'. h ⊢ Prog xa = Inr (r, h2)
⟶ h' ⊢ Prog xa = Inr (r', h2') ⟶ P h h' h2 h2' r r'"
and 4: "h ⊢ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r, h2)"
and 5: "h' ⊢ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r', h2')"
have h1:"⋀a. Complete_Partial_Order.chain (flat_ord (Inl e)) {y. ∃f∈A. y = f a}"
by (rule chain_fun[OF 1])
have "h ⊢ ?lub ∈ {y. ∃f∈A. y = f h}"
using flat_lub_in_chain[OF h1] 4
unfolding execute_def fun_lub_def
by (metis (mono_tags, lifting) Collect_cong Inl_Inr_False prog.sel)
moreover have "h' ⊢ ?lub ∈ {y. ∃f∈A. y = f h'}"
using flat_lub_in_chain[OF h1] 5
unfolding execute_def fun_lub_def
by (metis (no_types, lifting) Collect_cong Inl_Inr_False prog.sel)
ultimately obtain f where
"f ∈ A" and
"h ⊢ Prog f = Inr (r, h2)" and
"h' ⊢ Prog f = Inr (r', h2')"
using 1 4 5
apply(auto simp add: chain_def fun_ord_def flat_ord_def execute_def)[1]
by (metis Inl_Inr_False)
then show "P h h' h2 h2' r r'"
by(fact 2)
qed
definition dom_prog_ord ::
"('heap, exception, 'result) prog ⇒ ('heap, exception, 'result) prog ⇒ bool" where
"dom_prog_ord = img_ord (λa b. execute b a) (fun_ord (flat_ord (Inl NonTerminationException)))"
definition dom_prog_lub ::
"('heap, exception, 'result) prog set ⇒ ('heap, exception, 'result) prog" where
"dom_prog_lub = img_lub (λa b. execute b a) Prog (fun_lub (flat_lub (Inl NonTerminationException)))"
lemma dom_prog_lub_empty: "dom_prog_lub {} = error NonTerminationException"
by(simp add: dom_prog_lub_def img_lub_def fun_lub_def flat_lub_def error_def)
lemma dom_prog_interpretation: "partial_function_definitions dom_prog_ord dom_prog_lub"
proof -
have "partial_function_definitions (fun_ord (flat_ord (Inl NonTerminationException)))
(fun_lub (flat_lub (Inl NonTerminationException)))"
by (rule partial_function_lift) (rule flat_interpretation)
then show ?thesis
apply (simp add: dom_prog_lub_def dom_prog_ord_def flat_interpretation execute_def)
using partial_function_image prog.expand prog.sel by blast
qed
interpretation dom_prog: partial_function_definitions dom_prog_ord dom_prog_lub
rewrites "dom_prog_lub {} ≡ error NonTerminationException"
by (fact dom_prog_interpretation)(simp add: dom_prog_lub_empty)
lemma admissible_dom_prog:
"dom_prog.admissible (λf. ∀x h h' r. h ⊢ f x →⇩r r ⟶ h ⊢ f x →⇩h h' ⟶ P x h h' r)"
proof (rule admissible_fun[OF dom_prog_interpretation])
fix x
show "ccpo.admissible dom_prog_lub dom_prog_ord (λa. ∀h h' r. h ⊢ a →⇩r r ⟶ h ⊢ a →⇩h h'
⟶ P x h h' r)"
unfolding dom_prog_ord_def dom_prog_lub_def
proof (intro admissible_image partial_function_lift flat_interpretation)
show "ccpo.admissible (fun_lub (flat_lub (Inl NonTerminationException)))
(fun_ord (flat_ord (Inl NonTerminationException)))
((λa. ∀h h' r. h ⊢ a →⇩r r ⟶ h ⊢ a →⇩h h' ⟶ P x h h' r) ∘ Prog)"
by(auto simp add: execute_admissible returns_result_def returns_heap_def split: sum.splits)
next
show "⋀x y. (λb. b ⊢ x) = (λb. b ⊢ y) ⟹ x = y"
by(simp add: execute_def prog.expand)
next
show "⋀x. (λb. b ⊢ Prog x) = x"
by(simp add: execute_def)
qed
qed
lemma admissible_dom_prog2:
"dom_prog.admissible (λf. ∀x h h2 h' h2' r r2. h ⊢ f x →⇩r r ⟶ h ⊢ f x →⇩h h'
⟶ h2 ⊢ f x →⇩r r2 ⟶ h2 ⊢ f x →⇩h h2' ⟶ P x h h2 h' h2' r r2)"
proof (rule admissible_fun[OF dom_prog_interpretation])
fix x
show "ccpo.admissible dom_prog_lub dom_prog_ord (λa. ∀h h2 h' h2' r r2. h ⊢ a →⇩r r
⟶ h ⊢ a →⇩h h' ⟶ h2 ⊢ a →⇩r r2 ⟶ h2 ⊢ a →⇩h h2' ⟶ P x h h2 h' h2' r r2)"
unfolding dom_prog_ord_def dom_prog_lub_def
proof (intro admissible_image partial_function_lift flat_interpretation)
show "ccpo.admissible (fun_lub (flat_lub (Inl NonTerminationException)))
(fun_ord (flat_ord (Inl NonTerminationException)))
((λa. ∀h h2 h' h2' r r2. h ⊢ a →⇩r r ⟶ h ⊢ a →⇩h h' ⟶ h2 ⊢ a →⇩r r2 ⟶ h2 ⊢ a →⇩h h2'
⟶ P x h h2 h' h2' r r2) ∘ Prog)"
by(auto simp add: returns_result_def returns_heap_def intro!: ccpo.admissibleI
dest!: ccpo.admissibleD[OF execute_admissible2[where P="P x"]]
split: sum.splits)
next
show "⋀x y. (λb. b ⊢ x) = (λb. b ⊢ y) ⟹ x = y"
by(simp add: execute_def prog.expand)
next
show "⋀x. (λb. b ⊢ Prog x) = x"
by(simp add: execute_def)
qed
qed
lemma fixp_induct_dom_prog:
fixes F :: "'c ⇒ 'c" and
U :: "'c ⇒ 'b ⇒ ('heap, exception, 'result) prog" and
C :: "('b ⇒ ('heap, exception, 'result) prog) ⇒ 'c" and
P :: "'b ⇒ 'heap ⇒ 'heap ⇒ 'result ⇒ bool"
assumes mono: "⋀x. monotone (fun_ord dom_prog_ord) dom_prog_ord (λf. U (F (C f)) x)"
assumes eq: "f ≡ C (ccpo.fixp (fun_lub dom_prog_lub) (fun_ord dom_prog_ord) (λf. U (F (C f))))"
assumes inverse2: "⋀f. U (C f) = f"
assumes step: "⋀f x h h' r. (⋀x h h' r. h ⊢ (U f x) →⇩r r ⟹ h ⊢ (U f x) →⇩h h' ⟹ P x h h' r)
⟹ h ⊢ (U (F f) x) →⇩r r ⟹ h ⊢ (U (F f) x) →⇩h h' ⟹ P x h h' r"
assumes defined: "h ⊢ (U f x) →⇩r r" and "h ⊢ (U f x) →⇩h h'"
shows "P x h h' r"
using step defined dom_prog.fixp_induct_uc[of U F C, OF mono eq inverse2 admissible_dom_prog, of P]
by (metis assms(6) error_returns_heap)
declaration ‹Partial_Function.init "dom_prog" @{term dom_prog.fixp_fun}
@{term dom_prog.mono_body} @{thm dom_prog.fixp_rule_uc} @{thm dom_prog.fixp_induct_uc}
(SOME @{thm fixp_induct_dom_prog})›
abbreviation "mono_dom_prog ≡ monotone (fun_ord dom_prog_ord) dom_prog_ord"
lemma dom_prog_ordI:
assumes "⋀h. h ⊢ f →⇩e NonTerminationException ∨ h ⊢ f = h ⊢ g"
shows "dom_prog_ord f g"
proof(auto simp add: dom_prog_ord_def img_ord_def fun_ord_def flat_ord_def)[1]
fix x
assume "x ⊢ f ≠ x ⊢ g"
then show "x ⊢ f = Inl NonTerminationException"
using assms[where h=x]
by(auto simp add: returns_error_def split: sum.splits)
qed
lemma dom_prog_ordE:
assumes "dom_prog_ord x y"
obtains "h ⊢ x →⇩e NonTerminationException" | " h ⊢ x = h ⊢ y"
using assms unfolding dom_prog_ord_def img_ord_def fun_ord_def flat_ord_def
using returns_error_def by force
lemma bind_mono [partial_function_mono]:
fixes B :: "('a ⇒ ('heap, exception, 'result) prog) ⇒ ('heap, exception, 'result2) prog"
assumes mf: "mono_dom_prog B" and mg: "⋀y. mono_dom_prog (λf. C y f)"
shows "mono_dom_prog (λf. B f ⤜ (λy. C y f))"
proof (rule monotoneI)
fix f g :: "'a ⇒ ('heap, exception, 'result) prog"
assume fg: "dom_prog.le_fun f g"
from mf
have 1: "dom_prog_ord (B f) (B g)" by (rule monotoneD) (rule fg)
from mg
have 2: "⋀y'. dom_prog_ord (C y' f) (C y' g)" by (rule monotoneD) (rule fg)
have "dom_prog_ord (B f ⤜ (λy. C y f)) (B g ⤜ (λy. C y f))"
(is "dom_prog_ord ?L ?R")
proof (rule dom_prog_ordI)
fix h
from 1 show "h ⊢ ?L →⇩e NonTerminationException ∨ h ⊢ ?L = h ⊢ ?R"
apply(rule dom_prog_ordE)
apply(auto)[1]
using bind_cong by fastforce
qed
also
have h1: "dom_prog_ord (B g ⤜ (λy'. C y' f)) (B g ⤜ (λy'. C y' g))"
(is "dom_prog_ord ?L ?R")
proof (rule dom_prog_ordI)
fix h
show "h ⊢ ?L →⇩e NonTerminationException ∨ h ⊢ ?L = h ⊢ ?R"
proof (cases "h ⊢ ok (B g)")
case True
then obtain x h' where x: "h ⊢ B g →⇩r x" and h': "h ⊢ B g →⇩h h'"
by blast
then have "dom_prog_ord (C x f) (C x g)"
using 2 by simp
then show ?thesis
using x h'
apply(auto intro!: bind_returns_error_I3 dest: returns_result_eq dest!: dom_prog_ordE)[1]
apply(auto simp add: execute_bind_simp)[1]
using "2" dom_prog_ordE by metis
next
case False
then obtain e where e: "h ⊢ B g →⇩e e"
by(simp add: is_OK_def returns_error_def split: sum.splits)
have "h ⊢ B g ⤜ (λy'. C y' f) →⇩e e"
using e by(auto)
moreover have "h ⊢ B g ⤜ (λy'. C y' g) →⇩e e"
using e by auto
ultimately show ?thesis
using bind_returns_error_eq by metis
qed
qed
finally (dom_prog.leq_trans)
show "dom_prog_ord (B f ⤜ (λy. C y f)) (B g ⤜ (λy'. C y' g))" .
qed
lemma mono_dom_prog1 [partial_function_mono]:
fixes g :: "('a ⇒ ('heap, exception, 'result) prog) ⇒ 'b ⇒ ('heap, exception, 'result) prog"
assumes "⋀x. (mono_dom_prog (λf. g f x))"
shows "mono_dom_prog (λf. map_M (g f) xs)"
using assms
apply (induct xs)
by(auto simp add: call_mono dom_prog.const_mono intro!: bind_mono)
lemma mono_dom_prog2 [partial_function_mono]:
fixes g :: "('a ⇒ ('heap, exception, 'result) prog) ⇒ 'b ⇒ ('heap, exception, 'result) prog"
assumes "⋀x. (mono_dom_prog (λf. g f x))"
shows "mono_dom_prog (λf. forall_M (g f) xs)"
using assms
apply (induct xs)
by(auto simp add: call_mono dom_prog.const_mono intro!: bind_mono)
lemma sorted_list_set_cong [simp]:
"sorted_list_of_set (fset FS) = sorted_list_of_set (fset FS') ⟷ FS = FS'"
by auto
end
Theory ObjectPointer
section‹Object›
text‹In this theory, we introduce the typed pointer for the class Object. This class is the
common superclass of our class model.›
theory ObjectPointer
imports
Ref
begin
datatype 'object_ptr object_ptr = Ext 'object_ptr
register_default_tvars "'object_ptr object_ptr"
instantiation object_ptr :: (linorder) linorder
begin
definition less_eq_object_ptr :: "'object_ptr::linorder object_ptr ⇒ 'object_ptr object_ptr ⇒ bool"
where "less_eq_object_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j))"
definition less_object_ptr :: "'object_ptr::linorder object_ptr ⇒ 'object_ptr object_ptr ⇒ bool"
where "less_object_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance by(standard, auto simp add: less_eq_object_ptr_def less_object_ptr_def
split: object_ptr.splits)
end
end
Theory ObjectClass
section‹Object›
text‹In this theory, we introduce the definition of the class Object. This class is the
common superclass of our class model.›
theory ObjectClass
imports
BaseClass
"../pointers/ObjectPointer"
begin
record RObject =
nothing :: unit
register_default_tvars "'Object RObject_ext"
type_synonym 'Object Object = "'Object RObject_scheme"
register_default_tvars "'Object Object"
datatype ('object_ptr, 'Object) heap = Heap (the_heap: "((_) object_ptr, (_) Object) fmap")
register_default_tvars "('object_ptr, 'Object) heap"
type_synonym heap⇩f⇩i⇩n⇩a⇩l = "(unit, unit) heap"
definition object_ptr_kinds :: "(_) heap ⇒ (_) object_ptr fset"
where
"object_ptr_kinds = fmdom ∘ the_heap"
lemma object_ptr_kinds_simp [simp]:
"object_ptr_kinds (Heap (fmupd object_ptr object (the_heap h)))
= {|object_ptr|} |∪| object_ptr_kinds h"
by(auto simp add: object_ptr_kinds_def)
definition get⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) object_ptr ⇒ (_) heap ⇒ (_) Object option"
where
"get⇩O⇩b⇩j⇩e⇩c⇩t ptr h = fmlookup (the_heap h) ptr"
adhoc_overloading get get⇩O⇩b⇩j⇩e⇩c⇩t
locale l_type_wf_def⇩O⇩b⇩j⇩e⇩c⇩t
begin
definition a_type_wf :: "(_) heap ⇒ bool"
where
"a_type_wf h = True"
end
global_interpretation l_type_wf_def⇩O⇩b⇩j⇩e⇩c⇩t defines type_wf = a_type_wf .
lemmas type_wf_defs = a_type_wf_def
locale l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t = l_type_wf type_wf for type_wf :: "((_) heap ⇒ bool)" +
assumes type_wf⇩O⇩b⇩j⇩e⇩c⇩t: "type_wf h ⟹ ObjectClass.type_wf h"
locale l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas = l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t
begin
lemma get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf:
assumes "type_wf h"
shows "object_ptr |∈| object_ptr_kinds h ⟷ get⇩O⇩b⇩j⇩e⇩c⇩t object_ptr h ≠ None"
using l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t_axioms assms
apply(simp add: type_wf_def get⇩O⇩b⇩j⇩e⇩c⇩t_def)
by (simp add: fmlookup_dom_iff object_ptr_kinds_def)
end
global_interpretation l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas type_wf
by (simp add: l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas.intro l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t.intro)
definition put⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) object_ptr ⇒ (_) Object ⇒ (_) heap ⇒ (_) heap"
where
"put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h = Heap (fmupd ptr obj (the_heap h))"
adhoc_overloading put put⇩O⇩b⇩j⇩e⇩c⇩t
lemma put⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap:
assumes "put⇩O⇩b⇩j⇩e⇩c⇩t object_ptr object h = h'"
shows "object_ptr |∈| object_ptr_kinds h'"
using assms
unfolding put⇩O⇩b⇩j⇩e⇩c⇩t_def
by auto
lemma put⇩O⇩b⇩j⇩e⇩c⇩t_put_ptrs:
assumes "put⇩O⇩b⇩j⇩e⇩c⇩t object_ptr object h = h'"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|object_ptr|}"
using assms
by (metis comp_apply fmdom_fmupd funion_finsert_right heap.sel object_ptr_kinds_def
sup_bot.right_neutral put⇩O⇩b⇩j⇩e⇩c⇩t_def)
lemma object_more_extend_id [simp]: "more (extend x y) = y"
by(simp add: extend_def)
lemma object_empty [simp]: "⦇nothing = (), … = more x⦈ = x"
by simp
locale l_known_ptr⇩O⇩b⇩j⇩e⇩c⇩t
begin
definition a_known_ptr :: "(_) object_ptr ⇒ bool"
where
"a_known_ptr ptr = False"
lemma known_ptr_not_object_ptr:
"a_known_ptr ptr ⟹ ¬is_object_ptr ptr ⟹ known_ptr ptr"
by(simp add: a_known_ptr_def)
end
global_interpretation l_known_ptr⇩O⇩b⇩j⇩e⇩c⇩t defines known_ptr = a_known_ptr .
lemmas known_ptr_defs = a_known_ptr_def
locale l_known_ptrs = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool" +
fixes known_ptrs :: "(_) heap ⇒ bool"
assumes known_ptrs_known_ptr: "known_ptrs h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptr ptr"
assumes known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ known_ptrs h = known_ptrs h'"
assumes known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ known_ptrs h ⟹ known_ptrs h'"
assumes known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
known_ptrs h ⟹ known_ptrs h'"
locale l_known_ptrs⇩O⇩b⇩j⇩e⇩c⇩t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool"
begin
definition a_known_ptrs :: "(_) heap ⇒ bool"
where
"a_known_ptrs h = (∀ptr ∈ fset (object_ptr_kinds h). known_ptr ptr)"
lemma known_ptrs_known_ptr:
"a_known_ptrs h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptr ptr"
apply(simp add: a_known_ptrs_def)
using notin_fset by fastforce
lemma known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ a_known_ptrs h = a_known_ptrs h'"
by(auto simp add: a_known_ptrs_def)
lemma known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
lemma known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def)
end
global_interpretation l_known_ptrs⇩O⇩b⇩j⇩e⇩c⇩t known_ptr defines known_ptrs = a_known_ptrs .
lemmas known_ptrs_defs = a_known_ptrs_def
lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs"
using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset known_ptrs_new_ptr
by blast
lemma get_object_ptr_simp1 [simp]: "get⇩O⇩b⇩j⇩e⇩c⇩t object_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t object_ptr object h) = Some object"
by(simp add: get⇩O⇩b⇩j⇩e⇩c⇩t_def put⇩O⇩b⇩j⇩e⇩c⇩t_def)
lemma get_object_ptr_simp2 [simp]:
"object_ptr ≠ object_ptr'
⟹ get⇩O⇩b⇩j⇩e⇩c⇩t object_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t object_ptr' object h) = get⇩O⇩b⇩j⇩e⇩c⇩t object_ptr h"
by(simp add: get⇩O⇩b⇩j⇩e⇩c⇩t_def put⇩O⇩b⇩j⇩e⇩c⇩t_def)
subsection‹Limited Heap Modifications›
definition heap_unchanged_except :: "(_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
where
"heap_unchanged_except S h h' = (∀ptr ∈ (fset (object_ptr_kinds h)
∪ (fset (object_ptr_kinds h'))) - S. get ptr h = get ptr h')"
definition delete⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) object_ptr ⇒ (_) heap ⇒ (_) heap option" where
"delete⇩O⇩b⇩j⇩e⇩c⇩t ptr h = (if ptr |∈| object_ptr_kinds h then Some (Heap (fmdrop ptr (the_heap h)))
else None)"
lemma delete⇩O⇩b⇩j⇩e⇩c⇩t_pointer_removed:
assumes "delete⇩O⇩b⇩j⇩e⇩c⇩t ptr h = Some h'"
shows "ptr |∉| object_ptr_kinds h'"
using assms
by(auto simp add: delete⇩O⇩b⇩j⇩e⇩c⇩t_def object_ptr_kinds_def split: if_splits)
lemma delete⇩O⇩b⇩j⇩e⇩c⇩t_pointer_ptr_in_heap:
assumes "delete⇩O⇩b⇩j⇩e⇩c⇩t ptr h = Some h'"
shows "ptr |∈| object_ptr_kinds h"
using assms
by(auto simp add: delete⇩O⇩b⇩j⇩e⇩c⇩t_def object_ptr_kinds_def split: if_splits)
lemma delete⇩O⇩b⇩j⇩e⇩c⇩t_ok:
assumes "ptr |∈| object_ptr_kinds h"
shows "delete⇩O⇩b⇩j⇩e⇩c⇩t ptr h ≠ None"
using assms
by(auto simp add: delete⇩O⇩b⇩j⇩e⇩c⇩t_def object_ptr_kinds_def split: if_splits)
subsection ‹Code Generator Setup›
definition "create_heap xs = Heap (fmap_of_list xs)"
code_datatype ObjectClass.heap.Heap create_heap
lemma object_ptr_kinds_code3 [code]:
"fmlookup (the_heap (create_heap xs)) x = map_of xs x"
by(auto simp add: create_heap_def fmlookup_of_list)
lemma object_ptr_kinds_code4 [code]:
"the_heap (create_heap xs) = fmap_of_list xs"
by(simp add: create_heap_def)
lemma object_ptr_kinds_code5 [code]:
"the_heap (Heap x) = x"
by simp
end
Theory ObjectMonad
section‹Object›
text‹In this theory, we introduce the monadic method setup for the Object class.›
theory ObjectMonad
imports
BaseMonad
"../classes/ObjectClass"
begin
type_synonym ('object_ptr, 'Object, 'result) dom_prog
= "((_) heap, exception, 'result) prog"
register_default_tvars "('object_ptr, 'Object, 'result) dom_prog"
global_interpretation l_ptr_kinds_M object_ptr_kinds defines object_ptr_kinds_M = a_ptr_kinds_M .
lemmas object_ptr_kinds_M_defs = a_ptr_kinds_M_def
global_interpretation l_dummy defines get_M⇩O⇩b⇩j⇩e⇩c⇩t = "l_get_M.a_get_M get⇩O⇩b⇩j⇩e⇩c⇩t" .
lemma get_M_is_l_get_M: "l_get_M get⇩O⇩b⇩j⇩e⇩c⇩t type_wf object_ptr_kinds"
by (simp add: a_type_wf_def get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf l_get_M_def)
lemmas get_M_defs = get_M⇩O⇩b⇩j⇩e⇩c⇩t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]]
adhoc_overloading get_M get_M⇩O⇩b⇩j⇩e⇩c⇩t
locale l_get_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas = l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t
begin
interpretation l_get_M get⇩O⇩b⇩j⇩e⇩c⇩t type_wf object_ptr_kinds
apply(unfold_locales)
apply (simp add: get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf local.type_wf⇩O⇩b⇩j⇩e⇩c⇩t)
by (simp add: a_type_wf_def get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf)
lemmas get_M⇩O⇩b⇩j⇩e⇩c⇩t_ok = get_M_ok[folded get_M⇩O⇩b⇩j⇩e⇩c⇩t_def]
lemmas get_M⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap = get_M_ptr_in_heap[folded get_M⇩O⇩b⇩j⇩e⇩c⇩t_def]
end
global_interpretation l_get_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas type_wf
by (simp add: l_get_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas_def l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t_axioms)
lemma object_ptr_kinds_M_reads:
"reads (⋃object_ptr. {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing)}) object_ptr_kinds_M h h'"
apply(auto simp add: object_ptr_kinds_M_defs get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf type_wf_defs reads_def
preserved_def get_M_defs
split: option.splits)[1]
using a_type_wf_def get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf by blast+
global_interpretation l_put_M type_wf object_ptr_kinds get⇩O⇩b⇩j⇩e⇩c⇩t put⇩O⇩b⇩j⇩e⇩c⇩t
rewrites "a_get_M = get_M⇩O⇩b⇩j⇩e⇩c⇩t"
defines put_M⇩O⇩b⇩j⇩e⇩c⇩t = a_put_M
apply (simp add: get_M_is_l_get_M l_put_M_def)
by (simp add: get_M⇩O⇩b⇩j⇩e⇩c⇩t_def)
lemmas put_M_defs = a_put_M_def
adhoc_overloading put_M put_M⇩O⇩b⇩j⇩e⇩c⇩t
locale l_put_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas = l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t
begin
interpretation l_put_M type_wf object_ptr_kinds get⇩O⇩b⇩j⇩e⇩c⇩t put⇩O⇩b⇩j⇩e⇩c⇩t
apply(unfold_locales)
using get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t.type_wf⇩O⇩b⇩j⇩e⇩c⇩t local.l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t_axioms apply blast
by (simp add: a_type_wf_def get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf)
lemmas put_M⇩O⇩b⇩j⇩e⇩c⇩t_ok = put_M_ok[folded put_M⇩O⇩b⇩j⇩e⇩c⇩t_def]
lemmas put_M⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap = put_M_ptr_in_heap[folded put_M⇩O⇩b⇩j⇩e⇩c⇩t_def]
end
global_interpretation l_put_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas type_wf
by (simp add: l_put_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas_def l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t_axioms)
definition check_in_heap :: "(_) object_ptr ⇒ (_, unit) dom_prog"
where
"check_in_heap ptr = do {
h ← get_heap;
(if ptr |∈| object_ptr_kinds h then
return ()
else
error SegmentationFault
)}"
lemma check_in_heap_ptr_in_heap: "ptr |∈| object_ptr_kinds h ⟷ h ⊢ ok (check_in_heap ptr)"
by(auto simp add: check_in_heap_def)
lemma check_in_heap_pure [simp]: "pure (check_in_heap ptr) h"
by(auto simp add: check_in_heap_def intro!: bind_pure_I)
lemma check_in_heap_is_OK [simp]:
"ptr |∈| object_ptr_kinds h ⟹ h ⊢ ok (check_in_heap ptr ⤜ f) = h ⊢ ok (f ())"
by(simp add: check_in_heap_def)
lemma check_in_heap_returns_result [simp]:
"ptr |∈| object_ptr_kinds h ⟹ h ⊢ (check_in_heap ptr ⤜ f) →⇩r x = h ⊢ f () →⇩r x"
by(simp add: check_in_heap_def)
lemma check_in_heap_returns_heap [simp]:
"ptr |∈| object_ptr_kinds h ⟹ h ⊢ (check_in_heap ptr ⤜ f) →⇩h h' = h ⊢ f () →⇩h h'"
by(simp add: check_in_heap_def)
lemma check_in_heap_reads:
"reads {preserved (get_M object_ptr nothing)} (check_in_heap object_ptr) h h'"
apply(simp add: check_in_heap_def reads_def preserved_def)
by (metis a_type_wf_def get_M⇩O⇩b⇩j⇩e⇩c⇩t_ok get_M⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap is_OK_returns_result_E
is_OK_returns_result_I unit_all_impI)
subsection‹Invoke›
fun invoke_rec :: "(((_) object_ptr ⇒ bool) × ((_) object_ptr ⇒ 'args
⇒ (_, 'result) dom_prog)) list ⇒ (_) object_ptr ⇒ 'args
⇒ (_, 'result) dom_prog"
where
"invoke_rec ((P, f)#xs) ptr args = (if P ptr then f ptr args else invoke_rec xs ptr args)"
| "invoke_rec [] ptr args = error InvokeError"
definition invoke :: "(((_) object_ptr ⇒ bool) × ((_) object_ptr ⇒ 'args
⇒ (_, 'result) dom_prog)) list
⇒ (_) object_ptr ⇒ 'args ⇒ (_, 'result) dom_prog"
where
"invoke xs ptr args = do { check_in_heap ptr; invoke_rec xs ptr args}"
lemma invoke_split: "P (invoke ((Pred, f) # xs) ptr args) =
((¬(Pred ptr) ⟶ P (invoke xs ptr args))
∧ (Pred ptr ⟶ P (do {check_in_heap ptr; f ptr args})))"
by(simp add: invoke_def)
lemma invoke_split_asm: "P (invoke ((Pred, f) # xs) ptr args) =
(¬((¬(Pred ptr) ∧ (¬ P (invoke xs ptr args)))
∨ (Pred ptr ∧ (¬ P (do {check_in_heap ptr; f ptr args})))))"
by(simp add: invoke_def)
lemmas invoke_splits = invoke_split invoke_split_asm
lemma invoke_ptr_in_heap: "h ⊢ ok (invoke xs ptr args) ⟹ ptr |∈| object_ptr_kinds h"
by (metis bind_is_OK_E check_in_heap_ptr_in_heap invoke_def is_OK_returns_heap_I)
lemma invoke_pure [simp]: "pure (invoke [] ptr args) h"
by(auto simp add: invoke_def intro!: bind_pure_I)
lemma invoke_is_OK [simp]:
"ptr |∈| object_ptr_kinds h ⟹ Pred ptr
⟹ h ⊢ ok (invoke ((Pred, f) # xs) ptr args) = h ⊢ ok (f ptr args)"
by(simp add: invoke_def)
lemma invoke_returns_result [simp]:
"ptr |∈| object_ptr_kinds h ⟹ Pred ptr
⟹ h ⊢ (invoke ((Pred, f) # xs) ptr args) →⇩r x = h ⊢ f ptr args →⇩r x"
by(simp add: invoke_def)
lemma invoke_returns_heap [simp]:
"ptr |∈| object_ptr_kinds h ⟹ Pred ptr
⟹ h ⊢ (invoke ((Pred, f) # xs) ptr args) →⇩h h' = h ⊢ f ptr args →⇩h h'"
by(simp add: invoke_def)
lemma invoke_not [simp]: "¬Pred ptr ⟹ invoke ((Pred, f) # xs) ptr args = invoke xs ptr args"
by(auto simp add: invoke_def)
lemma invoke_empty [simp]: "¬h ⊢ ok (invoke [] ptr args)"
by(auto simp add: invoke_def check_in_heap_def)
lemma invoke_empty_reads [simp]: "∀P ∈ S. reflp P ∧ transp P ⟹ reads S (invoke [] ptr args) h h'"
apply(simp add: invoke_def reads_def preserved_def)
by (meson bind_returns_result_E error_returns_result)
subsection‹Modified Heaps›
lemma get_object_ptr_simp [simp]:
"get⇩O⇩b⇩j⇩e⇩c⇩t object_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h) = (if ptr = object_ptr then Some obj else get object_ptr h)"
by(auto simp add: get⇩O⇩b⇩j⇩e⇩c⇩t_def put⇩O⇩b⇩j⇩e⇩c⇩t_def split: option.splits Option.bind_splits)
lemma object_ptr_kinds_simp [simp]: "object_ptr_kinds (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h) = object_ptr_kinds h |∪| {|ptr|}"
by(auto simp add: object_ptr_kinds_def put⇩O⇩b⇩j⇩e⇩c⇩t_def split: option.splits)
lemma type_wf_put_I:
assumes "type_wf h"
shows "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
using assms
by(auto simp add: type_wf_defs split: option.splits)
lemma type_wf_put_ptr_not_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∉| object_ptr_kinds h"
shows "type_wf h"
using assms
by(auto simp add: type_wf_defs split: option.splits if_splits)
lemma type_wf_put_ptr_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∈| object_ptr_kinds h"
shows "type_wf h"
using assms
by(auto simp add: type_wf_defs split: option.splits if_splits)
subsection‹Preserving Types›
lemma type_wf_preserved: "type_wf h = type_wf h'"
by(auto simp add: type_wf_defs)
lemma object_ptr_kinds_preserved_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms
apply(auto simp add: object_ptr_kinds_def preserved_def get_M_defs get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: option.splits)[1]
apply (metis (mono_tags, lifting) domIff error_returns_result fmdom.rep_eq fmember.rep_eq
old.unit.exhaust option.case_eq_if return_returns_result)
by (metis (mono_tags, lifting) domIff error_returns_result fmdom.rep_eq fmember.rep_eq
old.unit.exhaust option.case_eq_if return_returns_result)
lemma object_ptr_kinds_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w object_ptr. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
proof -
{
fix object_ptr w
have "preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
apply(rule writes_small_big[OF assms])
by auto
}
then show ?thesis
using object_ptr_kinds_preserved_small by blast
qed
lemma reads_writes_preserved2:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' x. ∀w ∈ SW. h ⊢ w →⇩h h' ⟶ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr getter) h h'"
shows "preserved (get_M ptr getter) h h'"
apply(clarsimp simp add: preserved_def)
using reads_singleton assms(1) assms(2)
apply(rule reads_writes_preserved)
using assms(3)
by(auto simp add: preserved_def)
end
Theory NodePointer
section‹Node›
text‹In this theory, we introduce the typed pointers for the class Node.›
theory NodePointer
imports
ObjectPointer
begin
datatype 'node_ptr node_ptr = Ext 'node_ptr
register_default_tvars "'node_ptr node_ptr"
type_synonym ('object_ptr, 'node_ptr) object_ptr = "('node_ptr node_ptr + 'object_ptr) object_ptr"
register_default_tvars "('object_ptr, 'node_ptr) object_ptr"
definition cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ (_) object_ptr"
where
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr = object_ptr.Ext (Inl ptr)"
definition cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ (_) node_ptr option"
where
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r object_ptr = (case object_ptr of object_ptr.Ext (Inl node_ptr)
⇒ Some node_ptr | _ ⇒ None)"
adhoc_overloading cast cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r
definition is_node_ptr_kind :: "(_) object_ptr ⇒ bool"
where
"is_node_ptr_kind ptr = (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ≠ None)"
instantiation node_ptr :: (linorder) linorder
begin
definition less_eq_node_ptr :: "(_::linorder) node_ptr ⇒ (_) node_ptr ⇒ bool"
where "less_eq_node_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j))"
definition less_node_ptr :: "(_::linorder) node_ptr ⇒ (_) node_ptr ⇒ bool"
where "less_node_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance
apply(standard)
by(auto simp add: less_eq_node_ptr_def less_node_ptr_def split: node_ptr.splits)
end
lemma node_ptr_casts_commute [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = Some node_ptr ⟷ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr = ptr"
unfolding cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def
by(auto split: object_ptr.splits sum.splits)
lemma node_ptr_casts_commute2 [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr) = Some node_ptr"
by simp
lemma node_ptr_casts_commute3 [simp]:
assumes "is_node_ptr_kind ptr"
shows "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r (the (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr)) = ptr"
using assms
by(auto simp add: is_node_ptr_kind_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
split: object_ptr.splits sum.splits)
lemma is_node_ptr_kind_obtains:
assumes "is_node_ptr_kind ptr"
obtains node_ptr where "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = Some node_ptr"
using assms is_node_ptr_kind_def by auto
lemma is_node_ptr_kind_none:
assumes "¬is_node_ptr_kind ptr"
shows "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = None"
using assms
unfolding is_node_ptr_kind_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
by auto
lemma is_node_ptr_kind_cast [simp]: "is_node_ptr_kind (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)"
unfolding is_node_ptr_kind_def by simp
lemma cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_inject [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r x = cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r y ⟷ x = y"
by(simp add: cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def)
lemma cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_ext_none [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None"
by(simp add: cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def)
lemma node_ptr_inclusion [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr ∈ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ` node_ptrs ⟷ node_ptr ∈ node_ptrs"
by auto
end
Theory NodeClass
section‹Node›
text‹In this theory, we introduce the types for the Node class.›
theory NodeClass
imports
ObjectClass
"../pointers/NodePointer"
begin
subsubsection‹Node›
record RNode = RObject
+ nothing :: unit
register_default_tvars "'Node RNode_ext"
type_synonym 'Node Node = "'Node RNode_scheme"
register_default_tvars "'Node Node"
type_synonym ('Object, 'Node) Object = "('Node RNode_ext + 'Object) Object"
register_default_tvars "('Object, 'Node) Object"
type_synonym ('object_ptr, 'node_ptr, 'Object, 'Node) heap
= "('node_ptr node_ptr + 'object_ptr, 'Node RNode_ext + 'Object) heap"
register_default_tvars
"('object_ptr, 'node_ptr, 'Object, 'Node) heap"
type_synonym heap⇩f⇩i⇩n⇩a⇩l = "(unit, unit, unit, unit) heap"
definition node_ptr_kinds :: "(_) heap ⇒ (_) node_ptr fset"
where
"node_ptr_kinds heap =
(the |`| (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r |`| (ffilter is_node_ptr_kind (object_ptr_kinds heap))))"
lemma node_ptr_kinds_simp [simp]:
"node_ptr_kinds (Heap (fmupd (cast node_ptr) node (the_heap h)))
= {|node_ptr|} |∪| node_ptr_kinds h"
apply(auto simp add: node_ptr_kinds_def)[1]
by force
definition cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e :: "(_) Object ⇒ (_) Node option"
where
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e obj = (case RObject.more obj of Inl node
⇒ Some (RObject.extend (RObject.truncate obj) node) | _ ⇒ None)"
adhoc_overloading cast cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e
definition cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t:: "(_) Node ⇒ (_) Object"
where
"cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t node = (RObject.extend (RObject.truncate node) (Inl (RObject.more node)))"
adhoc_overloading cast cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t
definition is_node_kind :: "(_) Object ⇒ bool"
where
"is_node_kind ptr ⟷ cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e ptr ≠ None"
definition get⇩N⇩o⇩d⇩e :: "(_) node_ptr ⇒ (_) heap ⇒ (_) Node option"
where
"get⇩N⇩o⇩d⇩e node_ptr h = Option.bind (get (cast node_ptr) h) cast"
adhoc_overloading get get⇩N⇩o⇩d⇩e
locale l_type_wf_def⇩N⇩o⇩d⇩e
begin
definition a_type_wf :: "(_) heap ⇒ bool"
where
"a_type_wf h = (ObjectClass.type_wf h
∧ (∀node_ptr ∈ fset( node_ptr_kinds h). get⇩N⇩o⇩d⇩e node_ptr h ≠ None))"
end
global_interpretation l_type_wf_def⇩N⇩o⇩d⇩e defines type_wf = a_type_wf .
lemmas type_wf_defs = a_type_wf_def
locale l_type_wf⇩N⇩o⇩d⇩e = l_type_wf type_wf for type_wf :: "((_) heap ⇒ bool)" +
assumes type_wf⇩N⇩o⇩d⇩e: "type_wf h ⟹ NodeClass.type_wf h"
sublocale l_type_wf⇩N⇩o⇩d⇩e ⊆ l_type_wf⇩O⇩b⇩j⇩e⇩c⇩t
apply(unfold_locales)
using ObjectClass.a_type_wf_def by auto
locale l_get⇩N⇩o⇩d⇩e_lemmas = l_type_wf⇩N⇩o⇩d⇩e
begin
sublocale l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas by unfold_locales
lemma get⇩N⇩o⇩d⇩e_type_wf:
assumes "type_wf h"
shows "node_ptr |∈| node_ptr_kinds h ⟷ get⇩N⇩o⇩d⇩e node_ptr h ≠ None"
using l_type_wf⇩N⇩o⇩d⇩e_axioms assms
apply(simp add: type_wf_defs get⇩N⇩o⇩d⇩e_def l_type_wf⇩N⇩o⇩d⇩e_def)
by (metis bind_eq_None_conv ffmember_filter fimage_eqI fmember.rep_eq is_node_ptr_kind_cast
get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf node_ptr_casts_commute2 node_ptr_kinds_def option.sel option.simps(3))
end
global_interpretation l_get⇩N⇩o⇩d⇩e_lemmas type_wf
by unfold_locales
definition put⇩N⇩o⇩d⇩e :: "(_) node_ptr ⇒ (_) Node ⇒ (_) heap ⇒ (_) heap"
where
"put⇩N⇩o⇩d⇩e node_ptr node = put (cast node_ptr) (cast node)"
adhoc_overloading put put⇩N⇩o⇩d⇩e
lemma put⇩N⇩o⇩d⇩e_ptr_in_heap:
assumes "put⇩N⇩o⇩d⇩e node_ptr node h = h'"
shows "node_ptr |∈| node_ptr_kinds h'"
using assms
unfolding put⇩N⇩o⇩d⇩e_def node_ptr_kinds_def
by (metis ffmember_filter fimage_eqI is_node_ptr_kind_cast node_ptr_casts_commute2
option.sel put⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap)
lemma put⇩N⇩o⇩d⇩e_put_ptrs:
assumes "put⇩N⇩o⇩d⇩e node_ptr node h = h'"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast node_ptr|}"
using assms
by (simp add: put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_put_ptrs)
lemma node_ptr_kinds_commutes [simp]:
"cast node_ptr |∈| object_ptr_kinds h ⟷ node_ptr |∈| node_ptr_kinds h"
apply(auto simp add: node_ptr_kinds_def split: option.splits)[1]
by (metis (no_types, lifting) ffmember_filter fimage_eqI fset.map_comp
is_node_ptr_kind_none node_ptr_casts_commute2
option.distinct(1) option.sel)
lemma node_empty [simp]:
"⦇RObject.nothing = (), RNode.nothing = (), … = RNode.more node⦈ = node"
by simp
lemma cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t_inject [simp]: "cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t x = cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t y ⟷ x = y"
apply(simp add: cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def)
by (metis (full_types) RObject.surjective old.unit.exhaust)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_none [simp]:
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e obj = None ⟷ ¬ (∃node. cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t node = obj)"
apply(auto simp add: cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_def cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def split: sum.splits)[1]
by (metis (full_types) RObject.select_convs(2) RObject.surjective old.unit.exhaust)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_some [simp]: "cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e obj = Some node ⟷ cast node = obj"
by(auto simp add: cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_def cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def split: sum.splits)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv [simp]: "cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e (cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t node) = Some node"
by simp
locale l_known_ptr⇩N⇩o⇩d⇩e
begin
definition a_known_ptr :: "(_) object_ptr ⇒ bool"
where
"a_known_ptr ptr = False"
end
global_interpretation l_known_ptr⇩N⇩o⇩d⇩e defines known_ptr = a_known_ptr .
lemmas known_ptr_defs = a_known_ptr_def
locale l_known_ptrs⇩N⇩o⇩d⇩e = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool"
begin
definition a_known_ptrs :: "(_) heap ⇒ bool"
where
"a_known_ptrs h = (∀ptr ∈ fset (object_ptr_kinds h). known_ptr ptr)"
lemma known_ptrs_known_ptr: "a_known_ptrs h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptr ptr"
apply(simp add: a_known_ptrs_def)
using notin_fset by fastforce
lemma known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ a_known_ptrs h = a_known_ptrs h'"
by(auto simp add: a_known_ptrs_def)
lemma known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
lemma known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def)
end
global_interpretation l_known_ptrs⇩N⇩o⇩d⇩e known_ptr defines known_ptrs = a_known_ptrs .
lemmas known_ptrs_defs = a_known_ptrs_def
lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs"
using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset
known_ptrs_new_ptr
by blast
lemma get_node_ptr_simp1 [simp]: "get⇩N⇩o⇩d⇩e node_ptr (put⇩N⇩o⇩d⇩e node_ptr node h) = Some node"
by(auto simp add: get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def)
lemma get_node_ptr_simp2 [simp]:
"node_ptr ≠ node_ptr' ⟹ get⇩N⇩o⇩d⇩e node_ptr (put⇩N⇩o⇩d⇩e node_ptr' node h) = get⇩N⇩o⇩d⇩e node_ptr h"
by(auto simp add: get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def)
end
Theory NodeMonad
section‹Node›
text‹In this theory, we introduce the monadic method setup for the Node class.›
theory NodeMonad
imports
ObjectMonad
"../classes/NodeClass"
begin
type_synonym ('object_ptr, 'node_ptr, 'Object, 'Node, 'result) dom_prog
= "((_) heap, exception, 'result) prog"
register_default_tvars "('object_ptr, 'node_ptr, 'Object, 'Node, 'result) dom_prog"
global_interpretation l_ptr_kinds_M node_ptr_kinds defines node_ptr_kinds_M = a_ptr_kinds_M .
lemmas node_ptr_kinds_M_defs = a_ptr_kinds_M_def
lemma node_ptr_kinds_M_eq:
assumes "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
shows "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using assms
by(auto simp add: node_ptr_kinds_M_defs object_ptr_kinds_M_defs node_ptr_kinds_def)
global_interpretation l_dummy defines get_M⇩N⇩o⇩d⇩e = "l_get_M.a_get_M get⇩N⇩o⇩d⇩e" .
lemma get_M_is_l_get_M: "l_get_M get⇩N⇩o⇩d⇩e type_wf node_ptr_kinds"
apply(simp add: get⇩N⇩o⇩d⇩e_type_wf l_get_M_def)
by (metis ObjectClass.a_type_wf_def ObjectClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf bind_eq_None_conv get⇩N⇩o⇩d⇩e_def
node_ptr_kinds_commutes option.simps(3))
lemmas get_M_defs = get_M⇩N⇩o⇩d⇩e_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]]
adhoc_overloading get_M get_M⇩N⇩o⇩d⇩e
locale l_get_M⇩N⇩o⇩d⇩e_lemmas = l_type_wf⇩N⇩o⇩d⇩e
begin
sublocale l_get_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas by unfold_locales
interpretation l_get_M get⇩N⇩o⇩d⇩e type_wf node_ptr_kinds
apply(unfold_locales)
apply (simp add: get⇩N⇩o⇩d⇩e_type_wf local.type_wf⇩N⇩o⇩d⇩e)
by (meson NodeMonad.get_M_is_l_get_M l_get_M_def)
lemmas get_M⇩N⇩o⇩d⇩e_ok = get_M_ok[folded get_M⇩N⇩o⇩d⇩e_def]
end
global_interpretation l_get_M⇩N⇩o⇩d⇩e_lemmas type_wf by unfold_locales
lemma node_ptr_kinds_M_reads:
"reads (⋃object_ptr. {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing)}) node_ptr_kinds_M h h'"
using object_ptr_kinds_M_reads
apply (simp add: reads_def node_ptr_kinds_M_defs node_ptr_kinds_def
object_ptr_kinds_M_reads preserved_def)
by (smt object_ptr_kinds_preserved_small preserved_def unit_all_impI)
global_interpretation l_put_M type_wf node_ptr_kinds get⇩N⇩o⇩d⇩e put⇩N⇩o⇩d⇩e
rewrites "a_get_M = get_M⇩N⇩o⇩d⇩e"
defines put_M⇩N⇩o⇩d⇩e = a_put_M
apply (simp add: get_M_is_l_get_M l_put_M_def)
by (simp add: get_M⇩N⇩o⇩d⇩e_def)
lemmas put_M_defs = a_put_M_def
adhoc_overloading put_M put_M⇩N⇩o⇩d⇩e
locale l_put_M⇩N⇩o⇩d⇩e_lemmas = l_type_wf⇩N⇩o⇩d⇩e
begin
sublocale l_put_M⇩O⇩b⇩j⇩e⇩c⇩t_lemmas by unfold_locales
interpretation l_put_M type_wf node_ptr_kinds get⇩N⇩o⇩d⇩e put⇩N⇩o⇩d⇩e
apply(unfold_locales)
apply (simp add: get⇩N⇩o⇩d⇩e_type_wf local.type_wf⇩N⇩o⇩d⇩e)
by (meson NodeMonad.get_M_is_l_get_M l_get_M_def)
lemmas put_M⇩N⇩o⇩d⇩e_ok = put_M_ok[folded put_M⇩N⇩o⇩d⇩e_def]
end
global_interpretation l_put_M⇩N⇩o⇩d⇩e_lemmas type_wf by unfold_locales
lemma get_M_Object_preserved1 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x)) ⟹ h ⊢ put_M⇩N⇩o⇩d⇩e node_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast node_ptr = object_ptr")
by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs get⇩N⇩o⇩d⇩e_def preserved_def put⇩N⇩o⇩d⇩e_def
bind_eq_Some_conv
split: option.splits)
lemma get_M_Object_preserved2 [simp]:
"cast node_ptr ≠ object_ptr ⟹ h ⊢ put_M⇩N⇩o⇩d⇩e node_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def ObjectMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Object_preserved3 [simp]:
"h ⊢ put_M⇩N⇩o⇩d⇩e node_ptr setter v →⇩h h' ⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast node_ptr ≠ object_ptr")
by(auto simp add: put_M_defs get_M_defs get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def ObjectMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)
lemma get_M_Object_preserved4 [simp]:
"cast node_ptr ≠ object_ptr ⟹ h ⊢ put_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr setter v →⇩h h'
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
by(auto simp add: ObjectMonad.put_M_defs get_M_defs get⇩N⇩o⇩d⇩e_def ObjectMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
subsection‹Modified Heaps›
lemma get_node_ptr_simp [simp]:
"get⇩N⇩o⇩d⇩e node_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h) = (if ptr = cast node_ptr then cast obj else get node_ptr h)"
by(auto simp add: get⇩N⇩o⇩d⇩e_def)
lemma node_ptr_kinds_simp [simp]:
"node_ptr_kinds (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= node_ptr_kinds h |∪| (if is_node_ptr_kind ptr then {|the (cast ptr)|} else {||})"
by(auto simp add: node_ptr_kinds_def)
lemma type_wf_put_I:
assumes "type_wf h"
assumes "ObjectClass.type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "is_node_ptr_kind ptr ⟹ is_node_kind obj"
shows "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
using assms
apply(auto simp add: type_wf_defs split: option.splits)[1]
using cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_none is_node_kind_def apply blast
using cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_none is_node_kind_def apply blast
done
lemma type_wf_put_ptr_not_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∉| object_ptr_kinds h"
shows "type_wf h"
using assms
by(auto simp add: type_wf_defs elim!: ObjectMonad.type_wf_put_ptr_not_in_heap_E
split: option.splits if_splits)
lemma type_wf_put_ptr_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∈| object_ptr_kinds h"
assumes "ObjectClass.type_wf h"
assumes "is_node_ptr_kind ptr ⟹ is_node_kind (the (get ptr h))"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs split: option.splits if_splits)[1]
by (metis ObjectClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf bind.bind_lunit finite_set_in get⇩N⇩o⇩d⇩e_def is_node_kind_def option.exhaust_sel)
subsection‹Preserving Types›
lemma node_ptr_kinds_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "node_ptr_kinds h = node_ptr_kinds h'"
by(simp add: node_ptr_kinds_def preserved_def object_ptr_kinds_preserved_small[OF assms])
lemma node_ptr_kinds_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h'
⟶ (∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h')"
shows "node_ptr_kinds h = node_ptr_kinds h'"
using writes_small_big[OF assms]
apply(simp add: reflp_def transp_def preserved_def node_ptr_kinds_def)
by (metis assms object_ptr_kinds_preserved)
lemma type_wf_preserved_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
shows "type_wf h = type_wf h'"
using type_wf_preserved allI[OF assms(2), of id, simplified]
apply(auto simp add: type_wf_defs)[1]
apply(auto simp add: preserved_def get_M_defs node_ptr_kinds_small[OF assms(1)]
split: option.splits)[1]
apply (metis notin_fset option.simps(3))
by(auto simp add: preserved_def get_M_defs node_ptr_kinds_small[OF assms(1)]
split: option.splits, force)[1]
lemma type_wf_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
shows "type_wf h = type_wf h'"
proof -
have "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
using assms type_wf_preserved_small by fast
with assms(1) assms(2) show ?thesis
apply(rule writes_small_big)
by(auto simp add: reflp_def transp_def)
qed
end
Theory ElementPointer
section‹Element›
text‹In this theory, we introduce the typed pointers for the class Element.›
theory ElementPointer
imports
NodePointer
begin
datatype 'element_ptr element_ptr = Ref (the_ref: ref) | Ext 'element_ptr
register_default_tvars "'element_ptr element_ptr"
type_synonym ('node_ptr, 'element_ptr) node_ptr
= "('element_ptr element_ptr + 'node_ptr) node_ptr"
register_default_tvars "('node_ptr, 'element_ptr) node_ptr"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr) object_ptr
= "('object_ptr, 'element_ptr element_ptr + 'node_ptr) object_ptr"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr) object_ptr"
definition cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) element_ptr ⇒ (_) element_ptr"
where
"cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r = id"
definition cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) element_ptr ⇒ (_) node_ptr"
where
"cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = node_ptr.Ext (Inl ptr)"
abbreviation cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) element_ptr ⇒ (_) object_ptr"
where
"cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr)"
definition cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ (_) element_ptr option"
where
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r node_ptr = (case node_ptr of node_ptr.Ext (Inl element_ptr)
⇒ Some element_ptr | _ ⇒ None)"
abbreviation cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ (_) element_ptr option"
where
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr of
Some node_ptr ⇒ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r node_ptr
| None ⇒ None)"
adhoc_overloading cast cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r
cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r
consts is_element_ptr_kind :: 'a
definition is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ bool"
where
"is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = (case cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr of Some _ ⇒ True | _ ⇒ False)"
abbreviation is_element_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_element_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast ptr of
Some node_ptr ⇒ is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr
| None ⇒ False)"
adhoc_overloading is_element_ptr_kind is_element_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r
lemmas is_element_ptr_kind_def = is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
consts is_element_ptr :: 'a
definition is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) element_ptr ⇒ bool"
where
"is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr = (case ptr of element_ptr.Ref _ ⇒ True | _ ⇒ False)"
abbreviation is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ bool"
where
"is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ≡ (case cast ptr of
Some element_ptr ⇒ is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r element_ptr
| _ ⇒ False)"
abbreviation is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast ptr of
Some node_ptr ⇒ is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr
| None ⇒ False)"
adhoc_overloading is_element_ptr is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r
lemmas is_element_ptr_def = is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
consts is_element_ptr_ext :: 'a
abbreviation "is_element_ptr_ext⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr ≡ ¬ is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr"
abbreviation "is_element_ptr_ext⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ≡ is_element_ptr_kind ptr ∧ (¬ is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr)"
abbreviation "is_element_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ is_element_ptr_kind ptr ∧ (¬ is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr)"
adhoc_overloading is_element_ptr_ext is_element_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_element_ptr_ext⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r
instantiation element_ptr :: (linorder) linorder
begin
definition
less_eq_element_ptr :: "(_::linorder) element_ptr ⇒ (_)element_ptr ⇒ bool"
where
"less_eq_element_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j | Ref _ ⇒ False)
| Ref i ⇒ (case y of Ext _ ⇒ True | Ref j ⇒ i ≤ j))"
definition
less_element_ptr :: "(_::linorder) element_ptr ⇒ (_) element_ptr ⇒ bool"
where "less_element_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance
apply(standard)
by(auto simp add: less_eq_element_ptr_def less_element_ptr_def split: element_ptr.splits)
end
lemma is_element_ptr_ref [simp]: "is_element_ptr (element_ptr.Ref n)"
by(simp add: is_element_ptr⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def)
lemma element_ptr_casts_commute [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r node_ptr = Some element_ptr ⟷ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr = node_ptr"
unfolding cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
by(auto split: node_ptr.splits sum.splits)
lemma element_ptr_casts_commute2 [simp]:
"(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr) = Some element_ptr)"
by simp
lemma element_ptr_casts_commute3 [simp]:
assumes "is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr"
shows "cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r (the (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r node_ptr)) = node_ptr"
using assms
by(auto simp add: is_element_ptr_kind_def cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
split: node_ptr.splits sum.splits)
lemma is_element_ptr_kind_obtains:
assumes "is_element_ptr_kind node_ptr"
obtains element_ptr where "node_ptr = cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr"
by (metis assms is_element_ptr_kind_def case_optionE element_ptr_casts_commute)
lemma is_element_ptr_kind_none:
assumes "¬is_element_ptr_kind node_ptr"
shows "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r node_ptr = None"
using assms
unfolding is_element_ptr_kind_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
by(auto split: node_ptr.splits sum.splits)
lemma is_element_ptr_kind_cast [simp]:
"is_element_ptr_kind (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr)"
by (metis element_ptr_casts_commute is_element_ptr_kind_none option.distinct(1))
lemma cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_inject [simp]:
"cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r x = cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r y ⟷ x = y"
by(simp add: cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def)
lemma cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_ext_none [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r (node_ptr.Ext (Inr (Inr node_ext_ptr))) = None"
by(simp add: cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def)
lemma is_element_ptr_implies_kind [dest]: "is_element_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ⟹ is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr"
by(auto split: option.splits)
end
Theory CharacterDataPointer
section‹CharacterData›
text‹In this theory, we introduce the typed pointers for the class CharacterData.›
theory CharacterDataPointer
imports
ElementPointer
begin
datatype 'character_data_ptr character_data_ptr = Ref (the_ref: ref) | Ext 'character_data_ptr
register_default_tvars "'character_data_ptr character_data_ptr"
type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr
= "('character_data_ptr character_data_ptr + 'node_ptr, 'element_ptr) node_ptr"
register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr
= "('object_ptr, 'character_data_ptr character_data_ptr + 'node_ptr, 'element_ptr) object_ptr"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr"
definition cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) character_data_ptr ⇒ (_) node_ptr"
where
"cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = node_ptr.Ext (Inr (Inl ptr))"
abbreviation cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) character_data_ptr ⇒ (_) object_ptr"
where
"cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r (cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr)"
definition cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ (_) character_data_ptr option"
where
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r node_ptr = (case node_ptr of
node_ptr.Ext (Inr (Inl character_data_ptr)) ⇒ Some character_data_ptr
| _ ⇒ None)"
abbreviation cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ (_) character_data_ptr option"
where
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr of
Some node_ptr ⇒ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r node_ptr
| None ⇒ None)"
adhoc_overloading cast cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r
cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r
consts is_character_data_ptr_kind :: 'a
definition is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ bool"
where
"is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr = (case cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr
of Some _ ⇒ True | _ ⇒ False)"
abbreviation is_character_data_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_character_data_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast ptr of
Some node_ptr ⇒ is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr
| None ⇒ False)"
adhoc_overloading is_character_data_ptr_kind is_character_data_ptr_kind⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r
is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r
lemmas is_character_data_ptr_kind_def = is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
consts is_character_data_ptr :: 'a
definition is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r :: "(_) character_data_ptr ⇒ bool"
where
"is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr = (case ptr
of character_data_ptr.Ref _ ⇒ True | _ ⇒ False)"
abbreviation is_character_data_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ bool"
where
"is_character_data_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ≡ (case cast ptr of
Some character_data_ptr ⇒ is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r character_data_ptr
| _ ⇒ False)"
abbreviation is_character_data_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_character_data_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr of
Some node_ptr ⇒ is_character_data_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr
| None ⇒ False)"
adhoc_overloading is_character_data_ptr
is_character_data_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_character_data_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r
lemmas is_character_data_ptr_def = is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
consts is_character_data_ptr_ext :: 'a
abbreviation
"is_character_data_ptr_ext⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr ≡ ¬ is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr"
abbreviation "is_character_data_ptr_ext⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ptr of
Some character_data_ptr ⇒ is_character_data_ptr_ext⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r character_data_ptr
| None ⇒ False)"
abbreviation "is_character_data_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr of
Some node_ptr ⇒ is_character_data_ptr_ext⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr
| None ⇒ False)"
adhoc_overloading is_character_data_ptr_ext
is_character_data_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_character_data_ptr_ext⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r is_character_data_ptr_ext⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r
instantiation character_data_ptr :: (linorder) linorder
begin
definition
less_eq_character_data_ptr :: "(_::linorder) character_data_ptr ⇒ (_) character_data_ptr ⇒ bool"
where
"less_eq_character_data_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j | Ref _ ⇒ False)
| Ref i ⇒ (case y of Ext _ ⇒ True | Ref j ⇒ i ≤ j))"
definition
less_character_data_ptr :: "(_::linorder) character_data_ptr ⇒ (_) character_data_ptr ⇒ bool"
where "less_character_data_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance
apply(standard)
by(auto simp add: less_eq_character_data_ptr_def less_character_data_ptr_def
split: character_data_ptr.splits)
end
lemma is_character_data_ptr_ref [simp]: "is_character_data_ptr (character_data_ptr.Ref n)"
by(simp add: is_character_data_ptr⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def)
lemma cast_element_ptr_not_character_data_ptr [simp]:
"(cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr ≠ cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr)"
"(cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr ≠ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr)"
unfolding cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
by(auto)
lemma is_character_data_ptr_kind_not_element_ptr [simp]:
"¬ is_character_data_ptr_kind (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r element_ptr)"
unfolding is_character_data_ptr_kind_def cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
by auto
lemma is_element_ptr_kind_not_character_data_ptr [simp]:
"¬ is_element_ptr_kind (cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr)"
using is_element_ptr_kind_obtains by fastforce
lemma is_character_data_ptr_kind⇩_cast [simp]:
"is_character_data_ptr_kind (cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr)"
by (simp add: cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def)
lemma character_data_ptr_casts_commute [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r node_ptr = Some character_data_ptr
⟷ cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr = node_ptr"
unfolding cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
by(auto split: node_ptr.splits sum.splits)
lemma character_data_ptr_casts_commute2 [simp]:
"(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r (cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr) = Some character_data_ptr)"
by simp
lemma character_data_ptr_casts_commute3 [simp]:
assumes "is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr"
shows "cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r (the (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r node_ptr)) = node_ptr"
using assms
by(auto simp add: is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
split: node_ptr.splits sum.splits)
lemma is_character_data_ptr_kind_obtains:
assumes "is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr"
obtains character_data_ptr where "cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r character_data_ptr = node_ptr"
by (metis assms is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def case_optionE
character_data_ptr_casts_commute)
lemma is_character_data_ptr_kind_none:
assumes "¬is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr"
shows "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r node_ptr = None"
using assms
unfolding is_character_data_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
by(auto split: node_ptr.splits sum.splits)
lemma cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_inject [simp]:
"cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r x = cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r y ⟷ x = y"
by(simp add: cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def)
lemma cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_ext_none [simp]:
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r (node_ptr.Ext (Inr (Inr node_ext_ptr))) = None"
by(simp add: cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def)
end
Theory DocumentPointer
section‹Document›
text‹In this theory, we introduce the typed pointers for the class Document.›
theory DocumentPointer
imports
CharacterDataPointer
begin
datatype 'document_ptr document_ptr = Ref (the_ref: ref) | Ext 'document_ptr
register_default_tvars "'document_ptr document_ptr"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr) object_ptr
= "('document_ptr document_ptr + 'object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr) object_ptr"
definition cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_)document_ptr ⇒ (_) object_ptr"
where
"cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr = object_ptr.Ext (Inr (Inl ptr))"
definition cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ (_) document_ptr option"
where
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr = (case ptr of
object_ptr.Ext (Inr (Inl document_ptr)) ⇒ Some document_ptr
| _ ⇒ None)"
adhoc_overloading cast cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r
definition is_document_ptr_kind :: "(_) object_ptr ⇒ bool"
where
"is_document_ptr_kind ptr = (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr of
Some _ ⇒ True | None ⇒ False)"
consts is_document_ptr :: 'a
definition is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) document_ptr ⇒ bool"
where
"is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr = (case ptr of document_ptr.Ref _ ⇒ True | _ ⇒ False)"
abbreviation is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr of
Some document_ptr ⇒ is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr
| None ⇒ False)"
adhoc_overloading is_document_ptr is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r
lemmas is_document_ptr_def = is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
consts is_document_ptr_ext :: 'a
abbreviation "is_document_ptr_ext⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr ≡ ¬ is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr"
abbreviation "is_document_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr of
Some document_ptr ⇒ is_document_ptr_ext⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr
| None ⇒ False)"
adhoc_overloading is_document_ptr_ext is_document_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_document_ptr_ext⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r
instantiation document_ptr :: (linorder) linorder
begin
definition less_eq_document_ptr :: "(_::linorder) document_ptr ⇒ (_) document_ptr ⇒ bool"
where "less_eq_document_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j | Ref _ ⇒ False)
| Ref i ⇒ (case y of Ext _ ⇒ True | Ref j ⇒ i ≤ j))"
definition less_document_ptr :: "(_::linorder) document_ptr ⇒ (_) document_ptr ⇒ bool"
where "less_document_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance
apply(standard)
by(auto simp add: less_eq_document_ptr_def less_document_ptr_def split: document_ptr.splits)
end
lemma is_document_ptr_ref [simp]: "is_document_ptr (document_ptr.Ref n)"
by(simp add: is_document_ptr⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def)
lemma cast_document_ptr_not_node_ptr [simp]:
"cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr ≠ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr"
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr ≠ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr"
unfolding cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def
by auto
lemma document_ptr_no_node_ptr_cast [simp]:
"¬ is_document_ptr_kind (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)"
by(simp add: cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def is_document_ptr_kind_def)
lemma node_ptr_no_document_ptr_cast [simp]:
"¬ is_node_ptr_kind (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr)"
using is_node_ptr_kind_obtains by fastforce
lemma document_ptr_document_ptr_cast [simp]:
"is_document_ptr_kind (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr)"
by (simp add: cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def is_document_ptr_kind_def)
lemma document_ptr_casts_commute [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr = Some document_ptr ⟷ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr = ptr"
unfolding cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def
by(auto split: object_ptr.splits sum.splits)
lemma document_ptr_casts_commute2 [simp]:
"(cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr) = Some document_ptr)"
by simp
lemma document_ptr_casts_commute3 [simp]:
assumes "is_document_ptr_kind ptr"
shows "cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r (the (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr)) = ptr"
using assms
by(auto simp add: is_document_ptr_kind_def cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
split: object_ptr.splits sum.splits)
lemma is_document_ptr_kind_obtains:
assumes "is_document_ptr_kind ptr"
obtains document_ptr where "ptr = cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr"
using assms is_document_ptr_kind_def
by (metis case_optionE document_ptr_casts_commute)
lemma is_document_ptr_kind_none:
assumes "¬is_document_ptr_kind ptr"
shows "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr = None"
using assms
unfolding is_document_ptr_kind_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
by (auto split: object_ptr.splits sum.splits)
lemma cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_inject [simp]:
"cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r x = cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r y ⟷ x = y"
by(simp add: cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def)
lemma cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_ext_none [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None"
by(simp add: cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def)
lemma is_document_ptr_kind_not_element_ptr_kind [dest]:
"is_document_ptr_kind ptr ⟹ ¬ is_element_ptr_kind ptr"
by(auto simp add: split: option.splits)
end
Theory ShadowRootPointer
section‹ShadowRoot›
text‹In this theory, we introduce the typed pointers for the class ShadowRoot. Note that, in
this document, we will not make use of ShadowRoots nor will we discuss their particular properties.
We only include them here, as they are required for future work and they cannot be added alter
following the object-oriented extensibility of our data model.›
theory ShadowRootPointer
imports
"DocumentPointer"
begin
datatype 'shadow_root_ptr shadow_root_ptr = Ref (the_ref: ref) | Ext 'shadow_root_ptr
register_default_tvars "'shadow_root_ptr shadow_root_ptr"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr,
'document_ptr, 'shadow_root_ptr) object_ptr
= "('shadow_root_ptr shadow_root_ptr + 'object_ptr, 'node_ptr, 'element_ptr,
'character_data_ptr, 'document_ptr) object_ptr"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr,
'document_ptr, 'shadow_root_ptr) object_ptr"
definition cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r :: "(_) shadow_root_ptr ⇒ (_) shadow_root_ptr"
where
"cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r = id"
definition cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_)shadow_root_ptr ⇒ (_) object_ptr"
where
"cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr = object_ptr.Ext (Inr (Inr (Inl ptr)))"
definition cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ (_) shadow_root_ptr option"
where
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr = (case ptr of
object_ptr.Ext (Inr (Inr (Inl shadow_root_ptr))) ⇒ Some shadow_root_ptr
| _ ⇒ None)"
adhoc_overloading cast cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r
definition is_shadow_root_ptr_kind :: "(_) object_ptr ⇒ bool"
where
"is_shadow_root_ptr_kind ptr = (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr of Some _ ⇒ True
| None ⇒ False)"
consts is_shadow_root_ptr :: 'a
definition is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r :: "(_) shadow_root_ptr ⇒ bool"
where
"is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr = (case ptr of shadow_root_ptr.Ref _ ⇒ True
| _ ⇒ False)"
abbreviation is_shadow_root_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r :: "(_) object_ptr ⇒ bool"
where
"is_shadow_root_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr of
Some shadow_root_ptr ⇒ is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r shadow_root_ptr
| None ⇒ False)"
adhoc_overloading is_shadow_root_ptr is_shadow_root_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r
lemmas is_shadow_root_ptr_def = is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def
consts is_shadow_root_ptr_ext :: 'a
abbreviation "is_shadow_root_ptr_ext⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr ≡ ¬ is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr"
abbreviation "is_shadow_root_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr ≡ (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr of
Some shadow_root_ptr ⇒ is_shadow_root_ptr_ext⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r shadow_root_ptr
| None ⇒ False)"
adhoc_overloading is_shadow_root_ptr_ext is_shadow_root_ptr_ext⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r is_shadow_root_ptr_ext⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r
instantiation shadow_root_ptr :: (linorder) linorder
begin
definition
less_eq_shadow_root_ptr :: "(_::linorder) shadow_root_ptr ⇒ (_) shadow_root_ptr ⇒ bool"
where
"less_eq_shadow_root_ptr x y ≡ (case x of Ext i ⇒ (case y of Ext j ⇒ i ≤ j | Ref _ ⇒ False)
| Ref i ⇒ (case y of Ext _ ⇒ True | Ref j ⇒ i ≤ j))"
definition less_shadow_root_ptr :: "(_::linorder) shadow_root_ptr ⇒ (_) shadow_root_ptr ⇒ bool"
where "less_shadow_root_ptr x y ≡ x ≤ y ∧ ¬ y ≤ x"
instance
apply(standard)
by(auto simp add: less_eq_shadow_root_ptr_def less_shadow_root_ptr_def
split: shadow_root_ptr.splits)
end
lemma is_shadow_root_ptr_ref [simp]: "is_shadow_root_ptr (shadow_root_ptr.Ref n)"
by(simp add: is_shadow_root_ptr⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def)
lemma is_shadow_root_ptr_not_node_ptr[simp]: "¬is_shadow_root_ptr (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)"
by(simp add: is_shadow_root_ptr_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def)
lemma cast_shadow_root_ptr_not_node_ptr [simp]:
"cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr ≠ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr"
"cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr ≠ cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr"
unfolding cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def by auto
lemma cast_shadow_root_ptr_not_document_ptr [simp]:
"cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr ≠ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr"
"cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr ≠ cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr"
unfolding cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def by auto
lemma shadow_root_ptr_no_node_ptr_cast [simp]:
"¬ is_shadow_root_ptr_kind (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)"
by(simp add: cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def is_shadow_root_ptr_kind_def)
lemma node_ptr_no_shadow_root_ptr_cast [simp]:
"¬ is_node_ptr_kind (cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr)"
using is_node_ptr_kind_obtains by fastforce
lemma shadow_root_ptr_no_document_ptr_cast [simp]:
"¬ is_shadow_root_ptr_kind (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr)"
by(simp add: cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def is_shadow_root_ptr_kind_def)
lemma document_ptr_no_shadow_root_ptr_cast [simp]:
"¬ is_document_ptr_kind (cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr)"
using is_document_ptr_kind_obtains by fastforce
lemma shadow_root_ptr_shadow_root_ptr_cast [simp]:
"is_shadow_root_ptr_kind (cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr)"
by (simp add: cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def is_shadow_root_ptr_kind_def)
lemma shadow_root_ptr_casts_commute [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr = Some shadow_root_ptr ⟷ cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr = ptr"
unfolding cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def
by(auto split: object_ptr.splits sum.splits)
lemma shadow_root_ptr_casts_commute2 [simp]:
"(cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r (cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr) = Some shadow_root_ptr)"
by simp
lemma shadow_root_ptr_casts_commute3 [simp]:
assumes "is_shadow_root_ptr_kind ptr"
shows "cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r (the (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr)) = ptr"
using assms
by(auto simp add: is_shadow_root_ptr_kind_def cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def
split: object_ptr.splits sum.splits)
lemma is_shadow_root_ptr_kind_obtains:
assumes "is_shadow_root_ptr_kind ptr"
obtains shadow_root_ptr where "ptr = cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r shadow_root_ptr"
using assms is_shadow_root_ptr_kind_def
by (metis case_optionE shadow_root_ptr_casts_commute)
lemma is_shadow_root_ptr_kind_none:
assumes "¬is_shadow_root_ptr_kind ptr"
shows "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r ptr = None"
using assms
unfolding is_shadow_root_ptr_kind_def cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def
by (auto split: object_ptr.splits sum.splits)
lemma cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_inject [simp]:
"cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r x = cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r y ⟷ x = y"
by(simp add: cast⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_def)
lemma cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_ext_none [simp]:
"cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None"
by(simp add: cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩s⇩h⇩a⇩d⇩o⇩w⇩_⇩r⇩o⇩o⇩t⇩_⇩p⇩t⇩r_def)
lemma is_shadow_root_ptr_kind_simp1 [dest]: "is_document_ptr_kind ptr ⟹ ¬is_shadow_root_ptr_kind ptr"
by (metis document_ptr_no_shadow_root_ptr_cast shadow_root_ptr_casts_commute3)
lemma is_shadow_root_ptr_kind_simp2 [dest]: "is_node_ptr_kind ptr ⟹ ¬is_shadow_root_ptr_kind ptr"
by (metis node_ptr_no_shadow_root_ptr_cast shadow_root_ptr_casts_commute3)
end
Theory ElementClass
section‹Element›
text‹In this theory, we introduce the types for the Element class.›
theory ElementClass
imports
"NodeClass"
"ShadowRootPointer"
begin
text‹The type @{type "DOMString"} is a type synonym for @{type "string"}, define
in \autoref{sec:Core_DOM_Basic_Datatypes}.›
type_synonym attr_key = DOMString
type_synonym attr_value = DOMString
type_synonym attrs = "(attr_key, attr_value) fmap"
type_synonym tag_name = DOMString
record ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr) RElement = RNode +
nothing :: unit
tag_name :: tag_name
child_nodes :: "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr list"
attrs :: attrs
shadow_root_opt :: "'shadow_root_ptr shadow_root_ptr option"
type_synonym
('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element) Element
= "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option)
RElement_scheme"
register_default_tvars
"('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element) Element"
type_synonym
('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, 'Element) Node
= "(('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_ext
+ 'Node) Node"
register_default_tvars
"('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, 'Element) Node"
type_synonym
('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) Object
= "('Object, ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option)
RElement_ext + 'Node) Object"
register_default_tvars
"('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) Object"
type_synonym
('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr,
'Object, 'Node, 'Element) heap
= "('document_ptr document_ptr + 'shadow_root_ptr shadow_root_ptr + 'object_ptr,
'element_ptr element_ptr + 'character_data_ptr character_data_ptr + 'node_ptr, 'Object,
('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_ext +
'Node) heap"
register_default_tvars
"('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr,
'Object, 'Node, 'Element) heap"
type_synonym heap⇩f⇩i⇩n⇩a⇩l = "(unit, unit, unit, unit, unit, unit, unit, unit, unit) heap"
definition element_ptr_kinds :: "(_) heap ⇒ (_) element_ptr fset"
where
"element_ptr_kinds heap =
the |`| (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r |`| (ffilter is_element_ptr_kind (node_ptr_kinds heap)))"
lemma element_ptr_kinds_simp [simp]:
"element_ptr_kinds (Heap (fmupd (cast element_ptr) element (the_heap h))) =
{|element_ptr|} |∪| element_ptr_kinds h"
apply(auto simp add: element_ptr_kinds_def)[1]
by force
definition element_ptrs :: "(_) heap ⇒ (_) element_ptr fset"
where
"element_ptrs heap = ffilter is_element_ptr (element_ptr_kinds heap)"
definition cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t :: "(_) Node ⇒ (_) Element option"
where
"cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t node =
(case RNode.more node of Inl element ⇒ Some (RNode.extend (RNode.truncate node) element) | _ ⇒ None)"
adhoc_overloading cast cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t
abbreviation cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t :: "(_) Object ⇒ (_) Element option"
where
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t obj ≡ (case cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e obj of Some node ⇒ cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t node | None ⇒ None)"
adhoc_overloading cast cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t
definition cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e :: "(_) Element ⇒ (_) Node"
where
"cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element = RNode.extend (RNode.truncate element) (Inl (RNode.more element))"
adhoc_overloading cast cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e
abbreviation cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) Element ⇒ (_) Object"
where
"cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t ptr ≡ cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t (cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e ptr)"
adhoc_overloading cast cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t
consts is_element_kind :: 'a
definition is_element_kind⇩N⇩o⇩d⇩e :: "(_) Node ⇒ bool"
where
"is_element_kind⇩N⇩o⇩d⇩e ptr ⟷ cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr ≠ None"
adhoc_overloading is_element_kind is_element_kind⇩N⇩o⇩d⇩e
lemmas is_element_kind_def = is_element_kind⇩N⇩o⇩d⇩e_def
abbreviation is_element_kind⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) Object ⇒ bool"
where
"is_element_kind⇩O⇩b⇩j⇩e⇩c⇩t ptr ≡ cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr ≠ None"
adhoc_overloading is_element_kind is_element_kind⇩O⇩b⇩j⇩e⇩c⇩t
lemma element_ptr_kinds_commutes [simp]:
"cast element_ptr |∈| node_ptr_kinds h ⟷ element_ptr |∈| element_ptr_kinds h"
apply(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)[1]
by (metis (no_types, lifting) element_ptr_casts_commute2 ffmember_filter fimage_eqI
fset.map_comp is_element_ptr_kind_none node_ptr_casts_commute3
node_ptr_kinds_commutes node_ptr_kinds_def option.sel option.simps(3))
definition get⇩E⇩l⇩e⇩m⇩e⇩n⇩t :: "(_) element_ptr ⇒ (_) heap ⇒ (_) Element option"
where
"get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h = Option.bind (get⇩N⇩o⇩d⇩e (cast element_ptr) h) cast"
adhoc_overloading get get⇩E⇩l⇩e⇩m⇩e⇩n⇩t
locale l_type_wf_def⇩E⇩l⇩e⇩m⇩e⇩n⇩t
begin
definition a_type_wf :: "(_) heap ⇒ bool"
where
"a_type_wf h = (NodeClass.type_wf h ∧ (∀element_ptr ∈ fset (element_ptr_kinds h).
get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h ≠ None))"
end
global_interpretation l_type_wf_def⇩E⇩l⇩e⇩m⇩e⇩n⇩t defines type_wf = a_type_wf .
lemmas type_wf_defs = a_type_wf_def
locale l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t = l_type_wf type_wf for type_wf :: "((_) heap ⇒ bool)" +
assumes type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t: "type_wf h ⟹ ElementClass.type_wf h"
sublocale l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t ⊆ l_type_wf⇩N⇩o⇩d⇩e
apply(unfold_locales)
using NodeClass.a_type_wf_def
by (meson ElementClass.a_type_wf_def l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t_axioms l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
locale l_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
begin
sublocale l_get⇩N⇩o⇩d⇩e_lemmas by unfold_locales
lemma get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_type_wf:
assumes "type_wf h"
shows "element_ptr |∈| element_ptr_kinds h ⟷ get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h ≠ None"
using l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t_axioms assms
apply(simp add: type_wf_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
by (metis NodeClass.get⇩N⇩o⇩d⇩e_type_wf bind_eq_None_conv element_ptr_kinds_commutes notin_fset
option.distinct(1))
end
global_interpretation l_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas type_wf
by unfold_locales
definition put⇩E⇩l⇩e⇩m⇩e⇩n⇩t :: "(_) element_ptr ⇒ (_) Element ⇒ (_) heap ⇒ (_) heap"
where
"put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr element = put⇩N⇩o⇩d⇩e (cast element_ptr) (cast element)"
adhoc_overloading put put⇩E⇩l⇩e⇩m⇩e⇩n⇩t
lemma put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap:
assumes "put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr element h = h'"
shows "element_ptr |∈| element_ptr_kinds h'"
using assms
unfolding put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def element_ptr_kinds_def
by (metis element_ptr_kinds_commutes element_ptr_kinds_def put⇩N⇩o⇩d⇩e_ptr_in_heap)
lemma put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_put_ptrs:
assumes "put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr element h = h'"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast element_ptr|}"
using assms
by (simp add: put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_put_ptrs)
lemma cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e_inject [simp]:
"cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e x = cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e y ⟷ x = y"
apply(simp add: cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def)
by (metis (full_types) RNode.surjective old.unit.exhaust)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_none [simp]:
"cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t node = None ⟷ ¬ (∃element. cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element = node)"
apply(auto simp add: cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def
split: sum.splits)[1]
by (metis (full_types) RNode.select_convs(2) RNode.surjective old.unit.exhaust)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_some [simp]:
"cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t node = Some element ⟷ cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element = node"
by(auto simp add: cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def
split: sum.splits)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_inv [simp]: "cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t (cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element) = Some element"
by simp
lemma get_elment_ptr_simp1 [simp]:
"get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr (put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr element h) = Some element"
by(auto simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
lemma get_elment_ptr_simp2 [simp]:
"element_ptr ≠ element_ptr'
⟹ get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr (put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr' element h) = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h"
by(auto simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
abbreviation "create_element_obj tag_name_arg child_nodes_arg attrs_arg shadow_root_opt_arg
≡ ⦇ RObject.nothing = (), RNode.nothing = (), RElement.nothing = (),
tag_name = tag_name_arg, Element.child_nodes = child_nodes_arg, attrs = attrs_arg,
shadow_root_opt = shadow_root_opt_arg, … = None ⦈"
definition new⇩E⇩l⇩e⇩m⇩e⇩n⇩t :: "(_) heap ⇒ ((_) element_ptr × (_) heap)"
where
"new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h =
(let new_element_ptr = element_ptr.Ref (Suc (fMax (finsert 0 (element_ptr.the_ref
|`| (element_ptrs h)))))
in
(new_element_ptr, put new_element_ptr (create_element_obj '''' [] fmempty None) h))"
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "new_element_ptr |∈| element_ptr_kinds h'"
using assms
unfolding new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def
using put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap by blast
lemma new_element_ptr_new:
"element_ptr.Ref (Suc (fMax (finsert 0 (element_ptr.the_ref |`| element_ptrs h)))) |∉| element_ptrs h"
by (metis Suc_n_not_le_n element_ptr.sel(1) fMax_ge fimage_finsert finsertI1 finsertI2 set_finsert)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_not_in_heap:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "new_element_ptr |∉| element_ptr_kinds h"
using assms
unfolding new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
by (metis Pair_inject element_ptrs_def ffmember_filter new_element_ptr_new is_element_ptr_ref)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_new_ptr:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_element_ptr|}"
using assms
by (metis Pair_inject new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_put_ptrs)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_is_element_ptr:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "is_element_ptr new_element_ptr"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_get⇩O⇩b⇩j⇩e⇩c⇩t [simp]:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
assumes "ptr ≠ cast new_element_ptr"
shows "get⇩O⇩b⇩j⇩e⇩c⇩t ptr h = get⇩O⇩b⇩j⇩e⇩c⇩t ptr h'"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_get⇩N⇩o⇩d⇩e [simp]:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
assumes "ptr ≠ cast new_element_ptr"
shows "get⇩N⇩o⇩d⇩e ptr h = get⇩N⇩o⇩d⇩e ptr h'"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
assumes "ptr ≠ new_element_ptr"
shows "get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def)
locale l_known_ptr⇩E⇩l⇩e⇩m⇩e⇩n⇩t
begin
definition a_known_ptr :: "(_) object_ptr ⇒ bool"
where
"a_known_ptr ptr = (known_ptr ptr ∨ is_element_ptr ptr)"
lemma known_ptr_not_element_ptr: "¬is_element_ptr ptr ⟹ a_known_ptr ptr ⟹ known_ptr ptr"
by(simp add: a_known_ptr_def)
end
global_interpretation l_known_ptr⇩E⇩l⇩e⇩m⇩e⇩n⇩t defines known_ptr = a_known_ptr .
lemmas known_ptr_defs = a_known_ptr_def
locale l_known_ptrs⇩E⇩l⇩e⇩m⇩e⇩n⇩t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool"
begin
definition a_known_ptrs :: "(_) heap ⇒ bool"
where
"a_known_ptrs h = (∀ptr ∈ fset (object_ptr_kinds h). known_ptr ptr)"
lemma known_ptrs_known_ptr:
"ptr |∈| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ known_ptr ptr"
apply(simp add: a_known_ptrs_def)
using notin_fset by fastforce
lemma known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ a_known_ptrs h = a_known_ptrs h'"
by(auto simp add: a_known_ptrs_def)
lemma known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
lemma known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def)
end
global_interpretation l_known_ptrs⇩E⇩l⇩e⇩m⇩e⇩n⇩t known_ptr defines known_ptrs = a_known_ptrs .
lemmas known_ptrs_defs = a_known_ptrs_def
lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs"
using known_ptrs_known_ptr known_ptrs_preserved known_ptrs_subset known_ptrs_new_ptr l_known_ptrs_def by blast
end
Theory ElementMonad
section‹Element›
text‹In this theory, we introduce the monadic method setup for the Element class.›
theory ElementMonad
imports
NodeMonad
"ElementClass"
begin
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog
= "((_) heap, exception, 'result) prog"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr,
'document_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog"
global_interpretation l_ptr_kinds_M element_ptr_kinds defines element_ptr_kinds_M = a_ptr_kinds_M .
lemmas element_ptr_kinds_M_defs = a_ptr_kinds_M_def
lemma element_ptr_kinds_M_eq:
assumes "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
shows "|h ⊢ element_ptr_kinds_M|⇩r = |h' ⊢ element_ptr_kinds_M|⇩r"
using assms
by(auto simp add: element_ptr_kinds_M_defs node_ptr_kinds_M_defs element_ptr_kinds_def)
lemma element_ptr_kinds_M_reads:
"reads (⋃element_ptr. {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t element_ptr RObject.nothing)}) element_ptr_kinds_M h h'"
apply (simp add: reads_def node_ptr_kinds_M_defs element_ptr_kinds_M_defs element_ptr_kinds_def
node_ptr_kinds_M_reads preserved_def cong del: image_cong_simp)
apply (metis (mono_tags, hide_lams) node_ptr_kinds_small old.unit.exhaust preserved_def)
done
global_interpretation l_dummy defines get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t = "l_get_M.a_get_M get⇩E⇩l⇩e⇩m⇩e⇩n⇩t" .
lemma get_M_is_l_get_M: "l_get_M get⇩E⇩l⇩e⇩m⇩e⇩n⇩t type_wf element_ptr_kinds"
apply(simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_type_wf l_get_M_def)
by (metis (no_types, lifting) ObjectClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf ObjectClass.type_wf_defs
bind_eq_Some_conv bind_eq_Some_conv element_ptr_kinds_commutes get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def node_ptr_kinds_commutes option.simps(3))
lemmas get_M_defs = get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]]
adhoc_overloading get_M get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t
locale l_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
begin
sublocale l_get_M⇩N⇩o⇩d⇩e_lemmas by unfold_locales
interpretation l_get_M get⇩E⇩l⇩e⇩m⇩e⇩n⇩t type_wf element_ptr_kinds
apply(unfold_locales)
apply (simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_type_wf local.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t)
by (meson ElementMonad.get_M_is_l_get_M l_get_M_def)
lemmas get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok = get_M_ok[folded get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def]
lemmas get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap = get_M_ptr_in_heap[folded get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def]
end
global_interpretation l_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas type_wf by unfold_locales
global_interpretation l_put_M type_wf element_ptr_kinds get⇩E⇩l⇩e⇩m⇩e⇩n⇩t put⇩E⇩l⇩e⇩m⇩e⇩n⇩t
rewrites "a_get_M = get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t"
defines put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t = a_put_M
apply (simp add: get_M_is_l_get_M l_put_M_def)
by (simp add: get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
lemmas put_M_defs = a_put_M_def
adhoc_overloading put_M put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t
locale l_put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
begin
sublocale l_put_M⇩N⇩o⇩d⇩e_lemmas by unfold_locales
interpretation l_put_M type_wf element_ptr_kinds get⇩E⇩l⇩e⇩m⇩e⇩n⇩t put⇩E⇩l⇩e⇩m⇩e⇩n⇩t
apply(unfold_locales)
apply (simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_type_wf local.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t)
by (meson ElementMonad.get_M_is_l_get_M l_get_M_def)
lemmas put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok = put_M_ok[folded put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def]
end
global_interpretation l_put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas type_wf by unfold_locales
lemma element_put_get [simp]:
"h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h' ⟹ (⋀x. getter (setter (λ_. v) x) = v)
⟹ h' ⊢ get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr getter →⇩r v"
by(auto simp add: put_M_defs get_M_defs split: option.splits)
lemma get_M_Element_preserved1 [simp]:
"element_ptr ≠ element_ptr' ⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr' getter) h h'"
by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma element_put_get_preserved [simp]:
"(⋀x. getter (setter (λ_. v) x) = getter x) ⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr' getter) h h'"
apply(cases "element_ptr = element_ptr'")
by(auto simp add: put_M_defs get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Element_preserved3 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h' ⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast element_ptr = object_ptr")
by (auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def preserved_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def bind_eq_Some_conv
split: option.splits)
lemma get_M_Element_preserved4 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h' ⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
apply(cases "cast element_ptr = node_ptr")
by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def preserved_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def bind_eq_Some_conv
split: option.splits)
lemma get_M_Element_preserved5 [simp]:
"cast element_ptr ≠ node_ptr ⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def NodeMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Element_preserved6 [simp]:
"h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
apply(cases "cast element_ptr ≠ node_ptr")
by(auto simp add: put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def NodeMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)
lemma get_M_Element_preserved7 [simp]:
"cast element_ptr ≠ node_ptr ⟹ h ⊢ put_M⇩N⇩o⇩d⇩e node_ptr setter v →⇩h h'
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr getter) h h'"
by(auto simp add: NodeMonad.put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def NodeMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Element_preserved8 [simp]:
"cast element_ptr ≠ object_ptr ⟹ h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
ObjectMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Element_preserved9 [simp]:
"h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast element_ptr ≠ object_ptr")
by(auto simp add: put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def
ObjectMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)
lemma get_M_Element_preserved10 [simp]:
"cast element_ptr ≠ object_ptr ⟹ h ⊢ put_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr setter v →⇩h h'
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr getter) h h'"
by(auto simp add: ObjectMonad.put_M_defs get_M_defs get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def
ObjectMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
subsection‹Creating Elements›
definition new_element :: "(_, (_) element_ptr) dom_prog"
where
"new_element = do {
h ← get_heap;
(new_ptr, h') ← return (new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h);
return_heap h';
return new_ptr
}"
lemma new_element_ok [simp]:
"h ⊢ ok new_element"
by(auto simp add: new_element_def split: prod.splits)
lemma new_element_ptr_in_heap:
assumes "h ⊢ new_element →⇩h h'"
and "h ⊢ new_element →⇩r new_element_ptr"
shows "new_element_ptr |∈| element_ptr_kinds h'"
using assms
unfolding new_element_def
by(auto simp add: new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap is_OK_returns_result_I
elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_ptr_not_in_heap:
assumes "h ⊢ new_element →⇩h h'"
and "h ⊢ new_element →⇩r new_element_ptr"
shows "new_element_ptr |∉| element_ptr_kinds h"
using assms new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_not_in_heap
by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E
bind_returns_heap_E)
lemma new_element_new_ptr:
assumes "h ⊢ new_element →⇩h h'"
and "h ⊢ new_element →⇩r new_element_ptr"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_element_ptr|}"
using assms new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_new_ptr
by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E
bind_returns_heap_E)
lemma new_element_is_element_ptr:
assumes "h ⊢ new_element →⇩r new_element_ptr"
shows "is_element_ptr new_element_ptr"
using assms new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_is_element_ptr
by(auto simp add: new_element_def elim!: bind_returns_result_E split: prod.splits)
lemma new_element_child_nodes:
assumes "h ⊢ new_element →⇩h h'"
assumes "h ⊢ new_element →⇩r new_element_ptr"
shows "h' ⊢ get_M new_element_ptr child_nodes →⇩r []"
using assms
by(auto simp add: get_M_defs new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_tag_name:
assumes "h ⊢ new_element →⇩h h'"
assumes "h ⊢ new_element →⇩r new_element_ptr"
shows "h' ⊢ get_M new_element_ptr tag_name →⇩r ''''"
using assms
by(auto simp add: get_M_defs new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_attrs:
assumes "h ⊢ new_element →⇩h h'"
assumes "h ⊢ new_element →⇩r new_element_ptr"
shows "h' ⊢ get_M new_element_ptr attrs →⇩r fmempty"
using assms
by(auto simp add: get_M_defs new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_shadow_root_opt:
assumes "h ⊢ new_element →⇩h h'"
assumes "h ⊢ new_element →⇩r new_element_ptr"
shows "h' ⊢ get_M new_element_ptr shadow_root_opt →⇩r None"
using assms
by(auto simp add: get_M_defs new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_get_M⇩O⇩b⇩j⇩e⇩c⇩t:
"h ⊢ new_element →⇩h h' ⟹ h ⊢ new_element →⇩r new_element_ptr ⟹ ptr ≠ cast new_element_ptr
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr getter) h h'"
by(auto simp add: new_element_def ObjectMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_get_M⇩N⇩o⇩d⇩e:
"h ⊢ new_element →⇩h h' ⟹ h ⊢ new_element →⇩r new_element_ptr ⟹ ptr ≠ cast new_element_ptr
⟹ preserved (get_M⇩N⇩o⇩d⇩e ptr getter) h h'"
by(auto simp add: new_element_def NodeMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_element_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t:
"h ⊢ new_element →⇩h h' ⟹ h ⊢ new_element →⇩r new_element_ptr ⟹ ptr ≠ new_element_ptr
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_element_def get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
subsection‹Modified Heaps›
lemma get_Element_ptr_simp [simp]:
"get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= (if ptr = cast element_ptr then cast obj else get element_ptr h)"
by(auto simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def split: option.splits Option.bind_splits)
lemma element_ptr_kinds_simp [simp]:
"element_ptr_kinds (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= element_ptr_kinds h |∪| (if is_element_ptr_kind ptr then {|the (cast ptr)|} else {||})"
by(auto simp add: element_ptr_kinds_def is_node_ptr_kind_def split: option.splits)
lemma type_wf_put_I:
assumes "type_wf h"
assumes "NodeClass.type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "is_element_ptr_kind ptr ⟹ is_element_kind obj"
shows "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
using assms
by(auto simp add: type_wf_defs split: option.splits)
lemma type_wf_put_ptr_not_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∉| object_ptr_kinds h"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs elim!: NodeMonad.type_wf_put_ptr_not_in_heap_E
split: option.splits if_splits)[1]
using assms(2) node_ptr_kinds_commutes by blast
lemma type_wf_put_ptr_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∈| object_ptr_kinds h"
assumes "NodeClass.type_wf h"
assumes "is_element_ptr_kind ptr ⟹ is_element_kind (the (get ptr h))"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs split: option.splits if_splits)[1]
by (metis (no_types, lifting) NodeClass.l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas_axioms assms(2) bind.bind_lunit
cast⇩N⇩o⇩d⇩e⇩2⇩E⇩l⇩e⇩m⇩e⇩n⇩t_inv cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def
l_get⇩O⇩b⇩j⇩e⇩c⇩t_lemmas.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf option.collapse)
subsection‹Preserving Types›
lemma new_element_type_wf_preserved [simp]: "h ⊢ new_element →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
new_element_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: prod.splits if_splits elim!: bind_returns_heap_E)[1]
apply (metis element_ptr_kinds_commutes element_ptrs_def fempty_iff ffmember_filter finite_set_in
is_element_ptr_ref)
apply (metis element_ptrs_def fempty_iff ffmember_filter finite_set_in is_element_ptr_ref)
apply (metis (no_types, lifting) Suc_n_not_le_n element_ptr.sel(1) element_ptr_kinds_commutes
element_ptrs_def fMax_ge ffmember_filter fimage_eqI is_element_ptr_ref notin_fset)
apply (metis (no_types, lifting) Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def
fMax_ge ffmember_filter fimage_eqI finite_set_in is_element_ptr_ref)
done
locale l_new_element = l_type_wf +
assumes new_element_types_preserved: "h ⊢ new_element →⇩h h' ⟹ type_wf h = type_wf h'"
lemma new_element_is_l_new_element: "l_new_element type_wf"
using l_new_element.intro new_element_type_wf_preserved
by blast
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_tag_name_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr tag_name_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs
Let_def put_M_defs get_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def
get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1]
apply (metis finite_set_in option.inject)
apply (metis cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv finite_set_in option.sel)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_child_nodes_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr child_nodes_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs
Let_def put_M_defs get_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def
get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1]
apply (metis finite_set_in option.inject)
apply (metis cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv finite_set_in option.sel)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_attrs_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr attrs_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs Let_def
put_M_defs get_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1]
apply (metis finite_set_in option.inject)
apply (metis cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv finite_set_in option.sel)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_shadow_root_opt_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr shadow_root_opt_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs
Let_def put_M_defs get_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def
get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def
split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1]
apply (metis finite_set_in option.inject)
apply (metis cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv finite_set_in option.sel)
done
lemma put_M_pointers_preserved:
assumes "h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms
apply(auto simp add: put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def put⇩O⇩b⇩j⇩e⇩c⇩t_def
elim!: bind_returns_heap_E2 dest!: get_heap_E)[1]
by (meson get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap is_OK_returns_result_I)
lemma element_ptr_kinds_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h'
⟶ (∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h')"
shows "element_ptr_kinds h = element_ptr_kinds h'"
using writes_small_big[OF assms]
apply(simp add: reflp_def transp_def preserved_def element_ptr_kinds_def)
by (metis assms node_ptr_kinds_preserved)
lemma element_ptr_kinds_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "element_ptr_kinds h = element_ptr_kinds h'"
by(simp add: element_ptr_kinds_def node_ptr_kinds_def preserved_def
object_ptr_kinds_preserved_small[OF assms])
lemma type_wf_preserved_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
shows "type_wf h = type_wf h'"
using type_wf_preserved_small[OF assms(1) assms(2)] allI[OF assms(3), of id, simplified]
apply(auto simp add: type_wf_defs )[1]
apply(auto simp add: preserved_def get_M_defs element_ptr_kinds_small[OF assms(1)]
split: option.splits,force)[1]
by(auto simp add: preserved_def get_M_defs element_ptr_kinds_small[OF assms(1)]
split: option.splits,force)
lemma type_wf_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
shows "type_wf h = type_wf h'"
proof -
have "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
using assms type_wf_preserved_small by fast
with assms(1) assms(2) show ?thesis
apply(rule writes_small_big)
by(auto simp add: reflp_def transp_def)
qed
lemma type_wf_drop: "type_wf h ⟹ type_wf (Heap (fmdrop ptr (the_heap h)))"
apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs
node_ptr_kinds_def object_ptr_kinds_def is_node_ptr_kind_def
get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def get⇩O⇩b⇩j⇩e⇩c⇩t_def)[1]
apply (metis (no_types, lifting) element_ptr_kinds_commutes finite_set_in fmdom_notD fmdom_notI
fmlookup_drop heap.sel node_ptr_kinds_commutes o_apply object_ptr_kinds_def)
by (metis element_ptr_kinds_commutes fmdom_notI fmdrop_lookup heap.sel node_ptr_kinds_commutes
o_apply object_ptr_kinds_def)
end
Theory CharacterDataClass
section‹CharacterData›
text‹In this theory, we introduce the types for the CharacterData class.›
theory CharacterDataClass
imports
ElementClass
begin
subsubsection‹CharacterData›
text‹The type @{type "DOMString"} is a type synonym for @{type "string"}, defined
\autoref{sec:Core_DOM_Basic_Datatypes}.›
record RCharacterData = RNode +
nothing :: unit
val :: DOMString
register_default_tvars "'CharacterData RCharacterData_ext"
type_synonym 'CharacterData CharacterData = "'CharacterData option RCharacterData_scheme"
register_default_tvars "'CharacterData CharacterData"
type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node,
'Element, 'CharacterData) Node
= "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr,
'CharacterData option RCharacterData_ext + 'Node, 'Element) Node"
register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node,
'Element, 'CharacterData) Node"
type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node,
'Element, 'CharacterData) Object
= "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object,
'CharacterData option RCharacterData_ext + 'Node,
'Element) Object"
register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object,
'Node, 'Element, 'CharacterData) Object"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData) heap
= "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr,
'Object, 'CharacterData option RCharacterData_ext + 'Node, 'Element) heap"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData) heap"
type_synonym heap⇩f⇩i⇩n⇩a⇩l = "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap"
definition character_data_ptr_kinds :: "(_) heap ⇒ (_) character_data_ptr fset"
where
"character_data_ptr_kinds heap = the |`| (cast |`| (ffilter is_character_data_ptr_kind
(node_ptr_kinds heap)))"
lemma character_data_ptr_kinds_simp [simp]:
"character_data_ptr_kinds (Heap (fmupd (cast character_data_ptr) character_data (the_heap h)))
= {|character_data_ptr|} |∪| character_data_ptr_kinds h"
apply(auto simp add: character_data_ptr_kinds_def)[1]
by force
definition character_data_ptrs :: "(_) heap ⇒ _ character_data_ptr fset"
where
"character_data_ptrs heap = ffilter is_character_data_ptr (character_data_ptr_kinds heap)"
abbreviation "character_data_ptr_exts heap ≡ character_data_ptr_kinds heap - character_data_ptrs heap"
definition cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a :: "(_) Node ⇒ (_) CharacterData option"
where
"cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a node = (case RNode.more node of
Inr (Inl character_data) ⇒ Some (RNode.extend (RNode.truncate node) character_data)
| _ ⇒ None)"
adhoc_overloading cast cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
abbreviation cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a :: "(_) Object ⇒ (_) CharacterData option"
where
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a obj ≡ (case cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e obj of Some node ⇒ cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a node
| None ⇒ None)"
adhoc_overloading cast cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
definition cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e :: "(_) CharacterData ⇒ (_) Node"
where
"cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data = RNode.extend (RNode.truncate character_data)
(Inr (Inl (RNode.more character_data)))"
adhoc_overloading cast cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e
abbreviation cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) CharacterData ⇒ (_) Object"
where
"cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩O⇩b⇩j⇩e⇩c⇩t ptr ≡ cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t (cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e ptr)"
adhoc_overloading cast cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩O⇩b⇩j⇩e⇩c⇩t
consts is_character_data_kind :: 'a
definition is_character_data_kind⇩N⇩o⇩d⇩e :: "(_) Node ⇒ bool"
where
"is_character_data_kind⇩N⇩o⇩d⇩e ptr ⟷ cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr ≠ None"
adhoc_overloading is_character_data_kind is_character_data_kind⇩N⇩o⇩d⇩e
lemmas is_character_data_kind_def = is_character_data_kind⇩N⇩o⇩d⇩e_def
abbreviation is_character_data_kind⇩O⇩b⇩j⇩e⇩c⇩t :: "(_) Object ⇒ bool"
where
"is_character_data_kind⇩O⇩b⇩j⇩e⇩c⇩t ptr ≡ cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr ≠ None"
adhoc_overloading is_character_data_kind is_character_data_kind⇩O⇩b⇩j⇩e⇩c⇩t
lemma character_data_ptr_kinds_commutes [simp]:
"cast character_data_ptr |∈| node_ptr_kinds h
⟷ character_data_ptr |∈| character_data_ptr_kinds h"
apply(auto simp add: character_data_ptr_kinds_def)[1]
by (metis character_data_ptr_casts_commute2 comp_eq_dest_lhs ffmember_filter fimage_eqI
is_character_data_ptr_kind_none
option.distinct(1) option.sel)
definition get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a :: "(_) character_data_ptr ⇒ (_) heap ⇒ (_) CharacterData option"
where
"get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr h = Option.bind (get⇩N⇩o⇩d⇩e (cast character_data_ptr) h) cast"
adhoc_overloading get get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
locale l_type_wf_def⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
begin
definition a_type_wf :: "(_) heap ⇒ bool"
where
"a_type_wf h = (ElementClass.type_wf h
∧ (∀character_data_ptr ∈ fset (character_data_ptr_kinds h).
get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr h ≠ None))"
end
global_interpretation l_type_wf_def⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a defines type_wf = a_type_wf .
lemmas type_wf_defs = a_type_wf_def
locale l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a = l_type_wf type_wf for type_wf :: "((_) heap ⇒ bool)" +
assumes type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a: "type_wf h ⟹ CharacterDataClass.type_wf h"
sublocale l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ⊆ l_type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
apply(unfold_locales)
using ElementClass.a_type_wf_def
by (meson CharacterDataClass.a_type_wf_def l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_axioms l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
locale l_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas = l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
begin
sublocale l_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas by unfold_locales
lemma get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_type_wf:
assumes "type_wf h"
shows "character_data_ptr |∈| character_data_ptr_kinds h
⟷ get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr h ≠ None"
using l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_axioms assms
apply(simp add: type_wf_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
by (metis assms bind.bind_lzero character_data_ptr_kinds_commutes fmember.rep_eq
local.get⇩N⇩o⇩d⇩e_type_wf option.exhaust option.simps(3))
end
global_interpretation l_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas type_wf
by unfold_locales
definition put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a :: "(_) character_data_ptr ⇒ (_) CharacterData ⇒ (_) heap ⇒ (_) heap"
where
"put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr character_data = put⇩N⇩o⇩d⇩e (cast character_data_ptr)
(cast character_data)"
adhoc_overloading put put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
lemma put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_in_heap:
assumes "put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr character_data h = h'"
shows "character_data_ptr |∈| character_data_ptr_kinds h'"
using assms put⇩N⇩o⇩d⇩e_ptr_in_heap
unfolding put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def character_data_ptr_kinds_def
by (metis character_data_ptr_kinds_commutes character_data_ptr_kinds_def put⇩N⇩o⇩d⇩e_ptr_in_heap)
lemma put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_put_ptrs:
assumes "put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr character_data h = h'"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast character_data_ptr|}"
using assms
by (simp add: put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_put_ptrs)
lemma cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e_inject [simp]: "cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e x = cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e y ⟷ x = y"
apply(simp add: cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def)
by (metis (full_types) RNode.surjective old.unit.exhaust)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_none [simp]:
"cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a node = None ⟷ ¬ (∃character_data. cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data = node)"
apply(auto simp add: cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def
split: sum.splits)[1]
by (metis (full_types) RNode.select_convs(2) RNode.surjective old.unit.exhaust)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_some [simp]:
"cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a node = Some character_data ⟷ cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data = node"
by(auto simp add: cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e_def RObject.extend_def RNode.extend_def
split: sum.splits)
lemma cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_inv [simp]:
"cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a (cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data) = Some character_data"
by simp
lemma cast_element_not_character_data [simp]:
"(cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element ≠ cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data)"
"(cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e character_data ≠ cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e element)"
by(auto simp add: cast⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a⇩2⇩N⇩o⇩d⇩e_def cast⇩E⇩l⇩e⇩m⇩e⇩n⇩t⇩2⇩N⇩o⇩d⇩e_def RNode.extend_def)
lemma get_CharacterData_simp1 [simp]:
"get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr (put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr character_data h)
= Some character_data"
by(auto simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
lemma get_CharacterData_simp2 [simp]:
"character_data_ptr ≠ character_data_ptr' ⟹ get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr
(put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr' character_data h) = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr h"
by(auto simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
lemma get_CharacterData_simp3 [simp]:
"get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr (put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr f h) = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h"
by(auto simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
lemma get_CharacterData_simp4 [simp]:
"get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a element_ptr (put⇩E⇩l⇩e⇩m⇩e⇩n⇩t character_data_ptr f h) = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a element_ptr h"
by(auto simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a [simp]:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h'"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def)
abbreviation "create_character_data_obj val_arg
≡ ⦇ RObject.nothing = (), RNode.nothing = (), RCharacterData.nothing = (), val = val_arg, … = None ⦈"
definition new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a :: "(_) heap ⇒ ((_) character_data_ptr × (_) heap)"
where
"new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h =
(let new_character_data_ptr = character_data_ptr.Ref (Suc (fMax (character_data_ptr.the_ref
|`| (character_data_ptrs h)))) in
(new_character_data_ptr, put new_character_data_ptr (create_character_data_obj '''') h))"
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_in_heap:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "new_character_data_ptr |∈| character_data_ptr_kinds h'"
using assms
unfolding new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def
using put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_in_heap by blast
lemma new_character_data_ptr_new:
"character_data_ptr.Ref (Suc (fMax (finsert 0 (character_data_ptr.the_ref |`| character_data_ptrs h))))
|∉| character_data_ptrs h"
by (metis Suc_n_not_le_n character_data_ptr.sel(1) fMax_ge fimage_finsert finsertI1
finsertI2 set_finsert)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_not_in_heap:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "new_character_data_ptr |∉| character_data_ptr_kinds h"
using assms
unfolding new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def
by (metis Pair_inject character_data_ptrs_def fMax_finsert fempty_iff ffmember_filter
fimage_is_fempty is_character_data_ptr_ref max_0L new_character_data_ptr_new)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_new_ptr:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using assms
by (metis Pair_inject new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_put_ptrs)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_is_character_data_ptr:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "is_character_data_ptr new_character_data_ptr"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_get⇩O⇩b⇩j⇩e⇩c⇩t [simp]:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
assumes "ptr ≠ cast new_character_data_ptr"
shows "get⇩O⇩b⇩j⇩e⇩c⇩t ptr h = get⇩O⇩b⇩j⇩e⇩c⇩t ptr h'"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_get⇩N⇩o⇩d⇩e [simp]:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
assumes "ptr ≠ cast new_character_data_ptr"
shows "get⇩N⇩o⇩d⇩e ptr h = get⇩N⇩o⇩d⇩e ptr h'"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a [simp]:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
assumes "ptr ≠ new_character_data_ptr"
shows "get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h'"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def)
locale l_known_ptr⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
begin
definition a_known_ptr :: "(_) object_ptr ⇒ bool"
where
"a_known_ptr ptr = (known_ptr ptr ∨ is_character_data_ptr ptr)"
lemma known_ptr_not_character_data_ptr:
"¬is_character_data_ptr ptr ⟹ a_known_ptr ptr ⟹ known_ptr ptr"
by(simp add: a_known_ptr_def)
end
global_interpretation l_known_ptr⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a defines known_ptr = a_known_ptr .
lemmas known_ptr_defs = a_known_ptr_def
locale l_known_ptrs⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool"
begin
definition a_known_ptrs :: "(_) heap ⇒ bool"
where
"a_known_ptrs h = (∀ptr ∈ fset (object_ptr_kinds h). known_ptr ptr)"
lemma known_ptrs_known_ptr: "a_known_ptrs h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptr ptr"
apply(simp add: a_known_ptrs_def)
using notin_fset by fastforce
lemma known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ a_known_ptrs h = a_known_ptrs h'"
by(auto simp add: a_known_ptrs_def)
lemma known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
lemma known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def)
end
global_interpretation l_known_ptrs⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a known_ptr defines known_ptrs = a_known_ptrs .
lemmas known_ptrs_defs = a_known_ptrs_def
lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs"
using known_ptrs_known_ptr known_ptrs_preserved known_ptrs_subset known_ptrs_new_ptr l_known_ptrs_def
by blast
end
Theory CharacterDataMonad
section‹CharacterData›
text‹In this theory, we introduce the monadic method setup for the CharacterData class.›
theory CharacterDataMonad
imports
ElementMonad
"../classes/CharacterDataClass"
begin
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'result) dom_prog
= "((_) heap, exception, 'result) prog"
register_default_tvars
"('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr,
'Object, 'Node, 'Element, 'CharacterData, 'result) dom_prog"
global_interpretation l_ptr_kinds_M character_data_ptr_kinds
defines character_data_ptr_kinds_M = a_ptr_kinds_M .
lemmas character_data_ptr_kinds_M_defs = a_ptr_kinds_M_def
lemma character_data_ptr_kinds_M_eq:
assumes "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
shows "|h ⊢ character_data_ptr_kinds_M|⇩r = |h' ⊢ character_data_ptr_kinds_M|⇩r"
using assms
by(auto simp add: character_data_ptr_kinds_M_defs node_ptr_kinds_M_defs
character_data_ptr_kinds_def)
lemma character_data_ptr_kinds_M_reads:
"reads (⋃node_ptr. {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t node_ptr RObject.nothing)}) character_data_ptr_kinds_M h h'"
using node_ptr_kinds_M_reads
apply (simp add: reads_def node_ptr_kinds_M_defs character_data_ptr_kinds_M_defs
character_data_ptr_kinds_def preserved_def)
by (smt node_ptr_kinds_small preserved_def unit_all_impI)
global_interpretation l_dummy defines get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a = "l_get_M.a_get_M get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a" .
lemma get_M_is_l_get_M: "l_get_M get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a type_wf character_data_ptr_kinds"
apply(simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_type_wf l_get_M_def)
by (metis (no_types, hide_lams) NodeMonad.get_M_is_l_get_M bind_eq_Some_conv
character_data_ptr_kinds_commutes get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def l_get_M_def option.distinct(1))
lemmas get_M_defs = get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]]
adhoc_overloading get_M get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
locale l_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas = l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
begin
sublocale l_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas by unfold_locales
interpretation l_get_M get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a type_wf character_data_ptr_kinds
apply(unfold_locales)
apply (simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_type_wf local.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a)
by (meson CharacterDataMonad.get_M_is_l_get_M l_get_M_def)
lemmas get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ok = get_M_ok[folded get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def]
end
global_interpretation l_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas type_wf by unfold_locales
global_interpretation l_put_M type_wf character_data_ptr_kinds get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
rewrites "a_get_M = get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a" defines put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a = a_put_M
apply (simp add: get_M_is_l_get_M l_put_M_def)
by (simp add: get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
lemmas put_M_defs = a_put_M_def
adhoc_overloading put_M put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
locale l_put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas = l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
begin
sublocale l_put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_lemmas by unfold_locales
interpretation l_put_M type_wf character_data_ptr_kinds get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
apply(unfold_locales)
using get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_type_wf l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a local.l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_axioms
apply blast
by (meson CharacterDataMonad.get_M_is_l_get_M l_get_M_def)
lemmas put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ok = put_M_ok[folded put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def]
end
global_interpretation l_put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas type_wf by unfold_locales
lemma CharacterData_simp1 [simp]:
"(⋀x. getter (setter (λ_. v) x) = v) ⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ h' ⊢ get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr getter →⇩r v"
by(auto simp add: put_M_defs get_M_defs split: option.splits)
lemma CharacterData_simp2 [simp]:
"character_data_ptr ≠ character_data_ptr'
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr' getter) h h'"
by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma CharacterData_simp3 [simp]: "
(⋀x. getter (setter (λ_. v) x) = getter x)
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr' getter) h h'"
apply(cases "character_data_ptr = character_data_ptr'")
by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma CharacterData_simp4 [simp]:
"h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr getter) h h'"
by(auto simp add: put_M_defs ElementMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma CharacterData_simp5 [simp]:
"h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr getter) h h'"
by(auto simp add: ElementMonad.put_M_defs get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma CharacterData_simp6 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply (cases "cast character_data_ptr = object_ptr")
by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs
get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def preserved_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
bind_eq_Some_conv split: option.splits)
lemma CharacterData_simp7 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
apply(cases "cast character_data_ptr = node_ptr")
by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs
get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def preserved_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
bind_eq_Some_conv split: option.splits)
lemma CharacterData_simp8 [simp]:
"cast character_data_ptr ≠ node_ptr
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def NodeMonad.get_M_defs
preserved_def split: option.splits dest: get_heap_E)
lemma CharacterData_simp9 [simp]:
"h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
apply(cases "cast character_data_ptr ≠ node_ptr")
by(auto simp add: put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def
NodeMonad.get_M_defs preserved_def split: option.splits bind_splits
dest: get_heap_E)
lemma CharacterData_simp10 [simp]:
"cast character_data_ptr ≠ node_ptr
⟹ h ⊢ put_M⇩N⇩o⇩d⇩e node_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr getter) h h'"
by(auto simp add: NodeMonad.put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def NodeMonad.get_M_defs
preserved_def split: option.splits dest: get_heap_E)
lemma CharacterData_simp11 [simp]:
"cast character_data_ptr ≠ object_ptr
⟹ h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def
ObjectMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma CharacterData_simp12 [simp]:
"h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast character_data_ptr ≠ object_ptr")
apply(auto simp add: put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def
get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def ObjectMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)[1]
by(auto simp add: put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def
get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def ObjectMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)[1]
lemma CharacterData_simp13 [simp]:
"cast character_data_ptr ≠ object_ptr ⟹ h ⊢ put_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr getter) h h'"
by(auto simp add: ObjectMonad.put_M_defs get_M_defs get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def put⇩N⇩o⇩d⇩e_def
ObjectMonad.get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma new_element_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a:
"h ⊢ new_element →⇩h h' ⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr getter) h h'"
by(auto simp add: new_element_def get_M_defs preserved_def split: prod.splits option.splits
elim!: bind_returns_result_E bind_returns_heap_E)
subsection‹Creating CharacterData›
definition new_character_data :: "(_, (_) character_data_ptr) dom_prog"
where
"new_character_data = do {
h ← get_heap;
(new_ptr, h') ← return (new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h);
return_heap h';
return new_ptr
}"
lemma new_character_data_ok [simp]:
"h ⊢ ok new_character_data"
by(auto simp add: new_character_data_def split: prod.splits)
lemma new_character_data_ptr_in_heap:
assumes "h ⊢ new_character_data →⇩h h'"
and "h ⊢ new_character_data →⇩r new_character_data_ptr"
shows "new_character_data_ptr |∈| character_data_ptr_kinds h'"
using assms
unfolding new_character_data_def
by(auto simp add: new_character_data_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_in_heap
is_OK_returns_result_I
elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_ptr_not_in_heap:
assumes "h ⊢ new_character_data →⇩h h'"
and "h ⊢ new_character_data →⇩r new_character_data_ptr"
shows "new_character_data_ptr |∉| character_data_ptr_kinds h"
using assms new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_not_in_heap
by(auto simp add: new_character_data_def split: prod.splits
elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_new_ptr:
assumes "h ⊢ new_character_data →⇩h h'"
and "h ⊢ new_character_data →⇩r new_character_data_ptr"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using assms new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_new_ptr
by(auto simp add: new_character_data_def split: prod.splits
elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_is_character_data_ptr:
assumes "h ⊢ new_character_data →⇩r new_character_data_ptr"
shows "is_character_data_ptr new_character_data_ptr"
using assms new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_is_character_data_ptr
by(auto simp add: new_character_data_def elim!: bind_returns_result_E split: prod.splits)
lemma new_character_data_child_nodes:
assumes "h ⊢ new_character_data →⇩h h'"
assumes "h ⊢ new_character_data →⇩r new_character_data_ptr"
shows "h' ⊢ get_M new_character_data_ptr val →⇩r ''''"
using assms
by(auto simp add: get_M_defs new_character_data_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_get_M⇩O⇩b⇩j⇩e⇩c⇩t:
"h ⊢ new_character_data →⇩h h' ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ ptr ≠ cast new_character_data_ptr ⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr getter) h h'"
by(auto simp add: new_character_data_def ObjectMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_get_M⇩N⇩o⇩d⇩e:
"h ⊢ new_character_data →⇩h h' ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ ptr ≠ cast new_character_data_ptr ⟹ preserved (get_M⇩N⇩o⇩d⇩e ptr getter) h h'"
by(auto simp add: new_character_data_def NodeMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t:
"h ⊢ new_character_data →⇩h h' ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_character_data_def ElementMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a:
"h ⊢ new_character_data →⇩h h' ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ ptr ≠ new_character_data_ptr ⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr getter) h h'"
by(auto simp add: new_character_data_def get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
subsection‹Modified Heaps›
lemma get_CharacterData_ptr_simp [simp]:
"get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= (if ptr = cast character_data_ptr then cast obj else get character_data_ptr h)"
by(auto simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def split: option.splits Option.bind_splits)
lemma Character_data_ptr_kinds_simp [simp]:
"character_data_ptr_kinds (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h) = character_data_ptr_kinds h |∪|
(if is_character_data_ptr_kind ptr then {|the (cast ptr)|} else {||})"
by(auto simp add: character_data_ptr_kinds_def is_node_ptr_kind_def split: option.splits)
lemma type_wf_put_I:
assumes "type_wf h"
assumes "ElementClass.type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "is_character_data_ptr_kind ptr ⟹ is_character_data_kind obj"
shows "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
using assms
by(auto simp add: type_wf_defs split: option.splits)
lemma type_wf_put_ptr_not_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∉| object_ptr_kinds h"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs elim!: ElementMonad.type_wf_put_ptr_not_in_heap_E
split: option.splits if_splits)[1]
using assms(2) node_ptr_kinds_commutes by blast
lemma type_wf_put_ptr_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∈| object_ptr_kinds h"
assumes "ElementClass.type_wf h"
assumes "is_character_data_ptr_kind ptr ⟹ is_character_data_kind (the (get ptr h))"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs split: option.splits if_splits)[1]
by (metis (no_types, lifting) ElementClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf assms(2) bind.bind_lunit
cast⇩N⇩o⇩d⇩e⇩2⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_inv cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩N⇩o⇩d⇩e_inv get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def notin_fset option.collapse)
subsection‹Preserving Types›
lemma new_element_type_wf_preserved [simp]:
assumes "h ⊢ new_element →⇩h h'"
shows "type_wf h = type_wf h'"
using assms
apply(auto simp add: new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E
intro!: type_wf_put_I split: if_splits)[1]
using CharacterDataClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t assms new_element_type_wf_preserved apply blast
using element_ptrs_def apply fastforce
using CharacterDataClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t assms new_element_type_wf_preserved apply blast
by (metis Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def fMax_ge ffmember_filter
fimage_eqI is_element_ptr_ref)
lemma new_element_is_l_new_element: "l_new_element type_wf"
using l_new_element.intro new_element_type_wf_preserved
by blast
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_tag_name_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr tag_name_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1]
using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast
apply (metis (no_types, lifting) bind_eq_Some_conv finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
apply (metis finite_set_in)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_child_nodes_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr child_nodes_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
dest!: get_heap_E elim!: bind_returns_heap_E2
intro!: type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs
split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs
split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs
split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs
split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs
split: option.splits)[1]
using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast
apply (metis (no_types, lifting) bind_eq_Some_conv finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
apply (metis finite_set_in)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_attrs_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr attrs_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast
apply (metis (no_types, lifting) bind_eq_Some_conv finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
apply (metis finite_set_in)
done
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_shadow_root_opt_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr shadow_root_opt_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs split: option.splits)[1]
using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast
apply (metis (no_types, lifting) bind_eq_Some_conv finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def)
apply (metis finite_set_in)
done
lemma new_character_data_type_wf_preserved [simp]:
"h ⊢ new_character_data →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: new_character_data_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E
intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I
split: if_splits)[1]
apply(simp_all add: type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs is_node_kind_def)
by (meson new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_not_in_heap)
locale l_new_character_data = l_type_wf +
assumes new_character_data_types_preserved: "h ⊢ new_character_data →⇩h h' ⟹ type_wf h = type_wf h'"
lemma new_character_data_is_l_new_character_data: "l_new_character_data type_wf"
using l_new_character_data.intro new_character_data_type_wf_preserved
by blast
lemma put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_val_type_wf_preserved [simp]:
"h ⊢ put_M character_data_ptr val_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: CharacterDataMonad.put_M_defs put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
CharacterDataClass.type_wf⇩N⇩o⇩d⇩e CharacterDataClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs CharacterDataMonad.get_M_defs
split: option.splits)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs CharacterDataMonad.get_M_defs
ObjectClass.a_type_wf_def
split: option.splits)[1]
apply (metis (no_types, lifting) bind_eq_Some_conv finite_set_in get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def)
apply (metis finite_set_in)
done
lemma character_data_ptr_kinds_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "character_data_ptr_kinds h = character_data_ptr_kinds h'"
by(simp add: character_data_ptr_kinds_def node_ptr_kinds_def preserved_def
object_ptr_kinds_preserved_small[OF assms])
lemma character_data_ptr_kinds_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h'
⟶ (∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h')"
shows "character_data_ptr_kinds h = character_data_ptr_kinds h'"
using writes_small_big[OF assms]
apply(simp add: reflp_def transp_def preserved_def character_data_ptr_kinds_def)
by (metis assms node_ptr_kinds_preserved)
lemma type_wf_preserved_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
assumes "⋀character_data_ptr. preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr
RCharacterData.nothing) h h'"
shows "type_wf h = type_wf h'"
using type_wf_preserved_small[OF assms(1) assms(2) assms(3)]
allI[OF assms(4), of id, simplified] character_data_ptr_kinds_small[OF assms(1)]
apply(auto simp add: type_wf_defs preserved_def get_M_defs character_data_ptr_kinds_small[OF assms(1)]
split: option.splits)[1]
apply(force)
by force
lemma type_wf_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀character_data_ptr. preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr
RCharacterData.nothing) h h'"
shows "type_wf h = type_wf h'"
proof -
have "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
using assms type_wf_preserved_small by fast
with assms(1) assms(2) show ?thesis
apply(rule writes_small_big)
by(auto simp add: reflp_def transp_def)
qed
lemma type_wf_drop: "type_wf h ⟹ type_wf (Heap (fmdrop ptr (the_heap h)))"
apply(auto simp add: type_wf_def ElementMonad.type_wf_drop
l_type_wf_def⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a.a_type_wf_def)[1]
using type_wf_drop
by (metis (no_types, lifting) ElementClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t ObjectClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf
character_data_ptr_kinds_commutes finite_set_in fmlookup_drop get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def
get⇩O⇩b⇩j⇩e⇩c⇩t_def node_ptr_kinds_commutes object_ptr_kinds_code5)
end
Theory DocumentClass
section‹Document›
text‹In this theory, we introduce the types for the Document class.›
theory DocumentClass
imports
CharacterDataClass
begin
text‹The type @{type "doctype"} is a type synonym for @{type "string"}, defined
in \autoref{sec:Core_DOM_Basic_Datatypes}.›
record ('node_ptr, 'element_ptr, 'character_data_ptr) RDocument = RObject +
nothing :: unit
doctype :: doctype
document_element :: "(_) element_ptr option"
disconnected_nodes :: "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr list"
type_synonym
('node_ptr, 'element_ptr, 'character_data_ptr, 'Document) Document
= "('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option) RDocument_scheme"
register_default_tvars
"('node_ptr, 'element_ptr, 'character_data_ptr, 'Document) Document"
type_synonym
('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node,
'Element, 'CharacterData, 'Document) Object
= "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr,
('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option)
RDocument_ext + 'Object, 'Node, 'Element, 'CharacterData) Object"
register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr,
'Object, 'Node, 'Element, 'CharacterData, 'Document) Object"
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document) heap
= "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr,
('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option) RDocument_ext + 'Object, 'Node,
'Element, 'CharacterData) heap"
register_default_tvars
"('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document) heap"
type_synonym heap⇩f⇩i⇩n⇩a⇩l = "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap"
definition document_ptr_kinds :: "(_) heap ⇒ (_) document_ptr fset"
where
"document_ptr_kinds heap = the |`| (cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r |`|
(ffilter is_document_ptr_kind (object_ptr_kinds heap)))"
definition document_ptrs :: "(_) heap ⇒ (_) document_ptr fset"
where
"document_ptrs heap = ffilter is_document_ptr (document_ptr_kinds heap)"
definition cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t :: "(_) Object ⇒ (_) Document option"
where
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t obj = (case RObject.more obj of
Inr (Inl document) ⇒ Some (RObject.extend (RObject.truncate obj) document)
| _ ⇒ None)"
adhoc_overloading cast cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
definition cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t:: "(_) Document ⇒ (_) Object"
where
"cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t document = (RObject.extend (RObject.truncate document)
(Inr (Inl (RObject.more document))))"
adhoc_overloading cast cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t
definition is_document_kind :: "(_) Object ⇒ bool"
where
"is_document_kind ptr ⟷ cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr ≠ None"
lemma document_ptr_kinds_simp [simp]:
"document_ptr_kinds (Heap (fmupd (cast document_ptr) document (the_heap h)))
= {|document_ptr|} |∪| document_ptr_kinds h"
apply(auto simp add: document_ptr_kinds_def)[1]
by force
lemma document_ptr_kinds_commutes [simp]:
"cast document_ptr |∈| object_ptr_kinds h ⟷ document_ptr |∈| document_ptr_kinds h"
apply(auto simp add: object_ptr_kinds_def document_ptr_kinds_def)[1]
by (metis (no_types, lifting) document_ptr_casts_commute2 document_ptr_document_ptr_cast
ffmember_filter fimage_eqI fset.map_comp option.sel)
definition get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t :: "(_) document_ptr ⇒ (_) heap ⇒ (_) Document option"
where
"get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h = Option.bind (get (cast document_ptr) h) cast"
adhoc_overloading get get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
locale l_type_wf_def⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
begin
definition a_type_wf :: "(_) heap ⇒ bool"
where
"a_type_wf h = (CharacterDataClass.type_wf h ∧
(∀document_ptr ∈ fset (document_ptr_kinds h). get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h ≠ None))"
end
global_interpretation l_type_wf_def⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t defines type_wf = a_type_wf .
lemmas type_wf_defs = a_type_wf_def
locale l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t = l_type_wf type_wf for type_wf :: "((_) heap ⇒ bool)" +
assumes type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t: "type_wf h ⟹ DocumentClass.type_wf h"
sublocale l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ⊆ l_type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
apply(unfold_locales)
by (metis (full_types) type_wf_defs l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_axioms l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
locale l_get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
begin
sublocale l_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas by unfold_locales
lemma get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_type_wf:
assumes "type_wf h"
shows "document_ptr |∈| document_ptr_kinds h ⟷ get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h ≠ None"
using l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_axioms assms
apply(simp add: type_wf_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
by (metis document_ptr_kinds_commutes fmember.rep_eq is_none_bind is_none_simps(1)
is_none_simps(2) local.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf)
end
global_interpretation l_get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas type_wf by unfold_locales
definition put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t :: "(_) document_ptr ⇒ (_) Document ⇒ (_) heap ⇒ (_) heap"
where
"put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr document = put (cast document_ptr) (cast document)"
adhoc_overloading put put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
lemma put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_in_heap:
assumes "put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr document h = h'"
shows "document_ptr |∈| document_ptr_kinds h'"
using assms
unfolding put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
by (metis document_ptr_kinds_commutes put⇩O⇩b⇩j⇩e⇩c⇩t_ptr_in_heap)
lemma put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_put_ptrs:
assumes "put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr document h = h'"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast document_ptr|}"
using assms
by (simp add: put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩O⇩b⇩j⇩e⇩c⇩t_put_ptrs)
lemma cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t_inject [simp]: "cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t x = cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t y ⟷ x = y"
apply(simp add: cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def)
by (metis (full_types) RObject.surjective old.unit.exhaust)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_none [simp]:
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t obj = None ⟷ ¬ (∃document. cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t document = obj)"
apply(auto simp add: cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def
split: sum.splits)[1]
by (metis (full_types) RObject.select_convs(2) RObject.surjective old.unit.exhaust)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_some [simp]:
"cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t obj = Some document ⟷ cast document = obj"
by(auto simp add: cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def
split: sum.splits)
lemma cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_inv [simp]: "cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t (cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t document) = Some document"
by simp
lemma cast_document_not_node [simp]:
"cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t document ≠ cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t node"
"cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t node ≠ cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t document"
by(auto simp add: cast⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def cast⇩N⇩o⇩d⇩e⇩2⇩O⇩b⇩j⇩e⇩c⇩t_def RObject.extend_def)
lemma get_document_ptr_simp1 [simp]:
"get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr (put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr document h) = Some document"
by(auto simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemma get_document_ptr_simp2 [simp]:
"document_ptr ≠ document_ptr'
⟹ get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr (put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr' document h) = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h"
by(auto simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemma get_document_ptr_simp3 [simp]:
"get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr (put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr f h) = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr h"
by(auto simp add: get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemma get_document_ptr_simp4 [simp]:
"get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr (put⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr f h) = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h"
by(auto simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def)
lemma get_document_ptr_simp5 [simp]:
"get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr (put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr f h) = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr h"
by(auto simp add: get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def get⇩N⇩o⇩d⇩e_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemma get_document_ptr_simp6 [simp]:
"get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr (put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr f h) = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr h"
by(auto simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def)
lemma new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩E⇩l⇩e⇩m⇩e⇩n⇩t h = (new_element_ptr, h')"
shows "get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def)
lemma new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a h = (new_character_data_ptr, h')"
shows "get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def)
abbreviation
create_document_obj :: "char list ⇒ (_) element_ptr option ⇒ (_) node_ptr list ⇒ (_) Document"
where
"create_document_obj doctype_arg document_element_arg disconnected_nodes_arg
≡ ⦇ RObject.nothing = (), RDocument.nothing = (), doctype = doctype_arg,
document_element = document_element_arg,
disconnected_nodes = disconnected_nodes_arg, … = None ⦈"
definition new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t :: "(_)heap ⇒ ((_) document_ptr × (_) heap)"
where
"new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h =
(let new_document_ptr = document_ptr.Ref (Suc (fMax (finsert 0 (document_ptr.the_ref |`| (document_ptrs h)))))
in
(new_document_ptr, put new_document_ptr (create_document_obj '''' None []) h))"
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_in_heap:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "new_document_ptr |∈| document_ptr_kinds h'"
using assms
unfolding new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def
using put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_in_heap by blast
lemma new_document_ptr_new:
"document_ptr.Ref (Suc (fMax (finsert 0 (document_ptr.the_ref |`| document_ptrs h))))
|∉| document_ptrs h"
by (metis Suc_n_not_le_n document_ptr.sel(1) fMax_ge fimage_finsert finsertI1 finsertI2 set_finsert)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_not_in_heap:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "new_document_ptr |∉| document_ptr_kinds h"
using assms
unfolding new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
by (metis Pair_inject document_ptrs_def fMax_finsert fempty_iff ffmember_filter
fimage_is_fempty is_document_ptr_ref max_0L new_document_ptr_new)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_new_ptr:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_document_ptr|}"
using assms
by (metis Pair_inject new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_put_ptrs)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_is_document_ptr:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "is_document_ptr new_document_ptr"
using assms
by(auto simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_get⇩O⇩b⇩j⇩e⇩c⇩t [simp]:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
assumes "ptr ≠ cast new_document_ptr"
shows "get⇩O⇩b⇩j⇩e⇩c⇩t ptr h = get⇩O⇩b⇩j⇩e⇩c⇩t ptr h'"
using assms
by(auto simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_get⇩N⇩o⇩d⇩e [simp]:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "get⇩N⇩o⇩d⇩e ptr h = get⇩N⇩o⇩d⇩e ptr h'"
using assms
apply(simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
by(auto simp add: get⇩N⇩o⇩d⇩e_def)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_get⇩E⇩l⇩e⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h = get⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a [simp]:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
shows "get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h = get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr h'"
using assms
by(auto simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def)
lemma new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t [simp]:
assumes "new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h = (new_document_ptr, h')"
assumes "ptr ≠ new_document_ptr"
shows "get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h = get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr h'"
using assms
by(auto simp add: new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def)
locale l_known_ptr⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
begin
definition a_known_ptr :: "(_) object_ptr ⇒ bool"
where
"a_known_ptr ptr = (known_ptr ptr ∨ is_document_ptr ptr)"
lemma known_ptr_not_document_ptr: "¬is_document_ptr ptr ⟹ a_known_ptr ptr ⟹ known_ptr ptr"
by(simp add: a_known_ptr_def)
end
global_interpretation l_known_ptr⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t defines known_ptr = a_known_ptr .
lemmas known_ptr_defs = a_known_ptr_def
locale l_known_ptrs⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr ⇒ bool"
begin
definition a_known_ptrs :: "(_) heap ⇒ bool"
where
"a_known_ptrs h = (∀ptr ∈ fset (object_ptr_kinds h). known_ptr ptr)"
lemma known_ptrs_known_ptr: "a_known_ptrs h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptr ptr"
apply(simp add: a_known_ptrs_def)
using notin_fset by fastforce
lemma known_ptrs_preserved:
"object_ptr_kinds h = object_ptr_kinds h' ⟹ a_known_ptrs h = a_known_ptrs h'"
by(auto simp add: a_known_ptrs_def)
lemma known_ptrs_subset:
"object_ptr_kinds h' |⊆| object_ptr_kinds h ⟹ a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
lemma known_ptrs_new_ptr:
"object_ptr_kinds h' = object_ptr_kinds h |∪| {|new_ptr|} ⟹ known_ptr new_ptr ⟹
a_known_ptrs h ⟹ a_known_ptrs h'"
by(simp add: a_known_ptrs_def)
end
global_interpretation l_known_ptrs⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t known_ptr defines known_ptrs = a_known_ptrs .
lemmas known_ptrs_defs = a_known_ptrs_def
lemma known_ptrs_is_l_known_ptrs [instances]: "l_known_ptrs known_ptr known_ptrs"
using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset known_ptrs_new_ptr
by blast
end
Theory DocumentMonad
section‹Document›
text‹In this theory, we introduce the monadic method setup for the Document class.›
theory DocumentMonad
imports
CharacterDataMonad
"../classes/DocumentClass"
begin
type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document, 'result) dom_prog
= "((_) heap, exception, 'result) prog"
register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document, 'result) dom_prog"
global_interpretation l_ptr_kinds_M document_ptr_kinds defines document_ptr_kinds_M = a_ptr_kinds_M .
lemmas document_ptr_kinds_M_defs = a_ptr_kinds_M_def
lemma document_ptr_kinds_M_eq:
assumes "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
shows "|h ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using assms
by(auto simp add: document_ptr_kinds_M_defs object_ptr_kinds_M_defs document_ptr_kinds_def)
lemma document_ptr_kinds_M_reads:
"reads (⋃object_ptr. {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing)}) document_ptr_kinds_M h h'"
using object_ptr_kinds_M_reads
apply (simp add: reads_def object_ptr_kinds_M_defs document_ptr_kinds_M_defs
document_ptr_kinds_def preserved_def cong del: image_cong_simp)
apply (metis (mono_tags, hide_lams) object_ptr_kinds_preserved_small old.unit.exhaust preserved_def)
done
global_interpretation l_dummy defines get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t = "l_get_M.a_get_M get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t" .
lemma get_M_is_l_get_M: "l_get_M get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t type_wf document_ptr_kinds"
apply(simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_type_wf l_get_M_def)
by (metis ObjectClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf ObjectClass.type_wf_defs bind_eq_None_conv
document_ptr_kinds_commutes get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def option.simps(3))
lemmas get_M_defs = get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]]
adhoc_overloading get_M get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
locale l_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
begin
sublocale l_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas by unfold_locales
interpretation l_get_M get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t type_wf document_ptr_kinds
apply(unfold_locales)
apply (simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_type_wf local.type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t)
by (meson DocumentMonad.get_M_is_l_get_M l_get_M_def)
lemmas get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok = get_M_ok[folded get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def]
end
global_interpretation l_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas type_wf by unfold_locales
global_interpretation l_put_M type_wf document_ptr_kinds get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
rewrites "a_get_M = get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t" defines put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t = a_put_M
apply (simp add: get_M_is_l_get_M l_put_M_def)
by (simp add: get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def)
lemmas put_M_defs = a_put_M_def
adhoc_overloading put_M put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
locale l_put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas = l_type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
begin
sublocale l_put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_lemmas by unfold_locales
interpretation l_put_M type_wf document_ptr_kinds get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
apply(unfold_locales)
apply (simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_type_wf local.type_wf⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t)
by (meson DocumentMonad.get_M_is_l_get_M l_get_M_def)
lemmas put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok = put_M_ok[folded put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def]
end
global_interpretation l_put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_lemmas type_wf by unfold_locales
lemma document_put_get [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ (⋀x. getter (setter (λ_. v) x) = v)
⟹ h' ⊢ get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr getter →⇩r v"
by(auto simp add: put_M_defs get_M_defs split: option.splits)
lemma get_M_Mdocument_preserved1 [simp]:
"document_ptr ≠ document_ptr'
⟹ h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr' getter) h h'"
by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma document_put_get_preserved [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ (⋀x. getter (setter (λ_. v) x) = getter x)
⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr' getter) h h'"
apply(cases "document_ptr = document_ptr'")
by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved2 [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h' ⟹ preserved (get_M⇩N⇩o⇩d⇩e node_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs NodeMonad.get_M_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def get⇩N⇩o⇩d⇩e_def preserved_def split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved3 [simp]:
"cast document_ptr ≠ object_ptr
⟹ h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
by(auto simp add: put_M_defs get_M_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def ObjectMonad.get_M_defs
preserved_def split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved4 [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ (⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast document_ptr ≠ object_ptr")[1]
by(auto simp add: put_M_defs get_M_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
ObjectMonad.get_M_defs preserved_def
split: option.splits bind_splits dest: get_heap_E)
lemma get_M_Mdocument_preserved5 [simp]:
"cast document_ptr ≠ object_ptr
⟹ h ⊢ put_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr setter v →⇩h h'
⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr getter) h h'"
by(auto simp add: ObjectMonad.put_M_defs get_M_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def ObjectMonad.get_M_defs
preserved_def split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved6 [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h' ⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr getter) h h'"
by(auto simp add: put_M_defs ElementMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved7 [simp]:
"h ⊢ put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr setter v →⇩h h' ⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr getter) h h'"
by(auto simp add: ElementMonad.put_M_defs get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved8 [simp]:
"h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h'
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr getter) h h'"
by(auto simp add: put_M_defs CharacterDataMonad.get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved9 [simp]:
"h ⊢ put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr setter v →⇩h h'
⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr getter) h h'"
by(auto simp add: CharacterDataMonad.put_M_defs get_M_defs preserved_def
split: option.splits dest: get_heap_E)
lemma get_M_Mdocument_preserved10 [simp]:
"(⋀x. getter (cast (setter (λ_. v) x)) = getter (cast x))
⟹ h ⊢ put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr setter v →⇩h h' ⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr getter) h h'"
apply(cases "cast document_ptr = object_ptr")
by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
get⇩N⇩o⇩d⇩e_def preserved_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def bind_eq_Some_conv
split: option.splits)
lemma new_element_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t:
"h ⊢ new_element →⇩h h' ⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_element_def get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_character_data_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t:
"h ⊢ new_character_data →⇩h h' ⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_character_data_def get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
subsection ‹Creating Documents›
definition new_document :: "(_, (_) document_ptr) dom_prog"
where
"new_document = do {
h ← get_heap;
(new_ptr, h') ← return (new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t h);
return_heap h';
return new_ptr
}"
lemma new_document_ok [simp]:
"h ⊢ ok new_document"
by(auto simp add: new_document_def split: prod.splits)
lemma new_document_ptr_in_heap:
assumes "h ⊢ new_document →⇩h h'"
and "h ⊢ new_document →⇩r new_document_ptr"
shows "new_document_ptr |∈| document_ptr_kinds h'"
using assms
unfolding new_document_def
by(auto simp add: new_document_def new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_in_heap is_OK_returns_result_I
elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_ptr_not_in_heap:
assumes "h ⊢ new_document →⇩h h'"
and "h ⊢ new_document →⇩r new_document_ptr"
shows "new_document_ptr |∉| document_ptr_kinds h"
using assms new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ptr_not_in_heap
by(auto simp add: new_document_def split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_new_ptr:
assumes "h ⊢ new_document →⇩h h'"
and "h ⊢ new_document →⇩r new_document_ptr"
shows "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_document_ptr|}"
using assms new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_new_ptr
by(auto simp add: new_document_def split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_is_document_ptr:
assumes "h ⊢ new_document →⇩r new_document_ptr"
shows "is_document_ptr new_document_ptr"
using assms new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_is_document_ptr
by(auto simp add: new_document_def elim!: bind_returns_result_E split: prod.splits)
lemma new_document_doctype:
assumes "h ⊢ new_document →⇩h h'"
assumes "h ⊢ new_document →⇩r new_document_ptr"
shows "h' ⊢ get_M new_document_ptr doctype →⇩r ''''"
using assms
by(auto simp add: get_M_defs new_document_def new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_document_element:
assumes "h ⊢ new_document →⇩h h'"
assumes "h ⊢ new_document →⇩r new_document_ptr"
shows "h' ⊢ get_M new_document_ptr document_element →⇩r None"
using assms
by(auto simp add: get_M_defs new_document_def new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_disconnected_nodes:
assumes "h ⊢ new_document →⇩h h'"
assumes "h ⊢ new_document →⇩r new_document_ptr"
shows "h' ⊢ get_M new_document_ptr disconnected_nodes →⇩r []"
using assms
by(auto simp add: get_M_defs new_document_def new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def
split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_get_M⇩O⇩b⇩j⇩e⇩c⇩t:
"h ⊢ new_document →⇩h h' ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ ptr ≠ cast new_document_ptr ⟹ preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr getter) h h'"
by(auto simp add: new_document_def ObjectMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_get_M⇩N⇩o⇩d⇩e:
"h ⊢ new_document →⇩h h' ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ preserved (get_M⇩N⇩o⇩d⇩e ptr getter) h h'"
by(auto simp add: new_document_def NodeMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t:
"h ⊢ new_document →⇩h h' ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_document_def ElementMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a:
"h ⊢ new_document →⇩h h' ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ preserved (get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a ptr getter) h h'"
by(auto simp add: new_document_def CharacterDataMonad.get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
lemma new_document_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t:
"h ⊢ new_document →⇩h h'
⟹ h ⊢ new_document →⇩r new_document_ptr ⟹ ptr ≠ new_document_ptr
⟹ preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t ptr getter) h h'"
by(auto simp add: new_document_def get_M_defs preserved_def
split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E)
subsection ‹Modified Heaps›
lemma get_document_ptr_simp [simp]:
"get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= (if ptr = cast document_ptr then cast obj else get document_ptr h)"
by(auto simp add: get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def split: option.splits Option.bind_splits)
lemma document_ptr_kidns_simp [simp]:
"document_ptr_kinds (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)
= document_ptr_kinds h |∪| (if is_document_ptr_kind ptr then {|the (cast ptr)|} else {||})"
by(auto simp add: document_ptr_kinds_def split: option.splits)
lemma type_wf_put_I:
assumes "type_wf h"
assumes "CharacterDataClass.type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "is_document_ptr_kind ptr ⟹ is_document_kind obj"
shows "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
using assms
by(auto simp add: type_wf_defs is_document_kind_def split: option.splits)
lemma type_wf_put_ptr_not_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∉| object_ptr_kinds h"
shows "type_wf h"
using assms
by(auto simp add: type_wf_defs elim!: CharacterDataMonad.type_wf_put_ptr_not_in_heap_E
split: option.splits if_splits)
lemma type_wf_put_ptr_in_heap_E:
assumes "type_wf (put⇩O⇩b⇩j⇩e⇩c⇩t ptr obj h)"
assumes "ptr |∈| object_ptr_kinds h"
assumes "CharacterDataClass.type_wf h"
assumes "is_document_ptr_kind ptr ⟹ is_document_kind (the (get ptr h))"
shows "type_wf h"
using assms
apply(auto simp add: type_wf_defs elim!: CharacterDataMonad.type_wf_put_ptr_in_heap_E
split: option.splits if_splits)[1]
by (metis (no_types, lifting) CharacterDataClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf bind.bind_lunit get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
is_document_kind_def notin_fset option.exhaust_sel)
subsection ‹Preserving Types›
lemma new_element_type_wf_preserved [simp]:
"h ⊢ new_element →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: new_element_def new⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def Let_def put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def element_ptrs_def
elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I
split: if_splits)[1]
apply fastforce
by (metis Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def fMax_ge ffmember_filter
fimage_eqI is_element_ptr_ref)
lemma new_element_is_l_new_element [instances]:
"l_new_element type_wf"
using l_new_element.intro new_element_type_wf_preserved
by blast
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_tag_name_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr tag_name_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs
ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply (metis NodeClass.a_type_wf_def NodeClass.get⇩N⇩o⇩d⇩e_type_wf ObjectClass.a_type_wf_def
bind.bind_lzero finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def l_type_wf_def⇩N⇩o⇩d⇩e.a_type_wf_def option.collapse
option.distinct(1) option.simps(3))
by (metis fmember.rep_eq)
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_child_nodes_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr child_nodes_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply (metis NodeClass.a_type_wf_def NodeClass.get⇩N⇩o⇩d⇩e_type_wf ObjectClass.a_type_wf_def
bind.bind_lzero finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def l_type_wf_def⇩N⇩o⇩d⇩e.a_type_wf_def option.collapse
option.distinct(1) option.simps(3))
by (metis fmember.rep_eq)
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_attrs_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr attrs_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply (metis NodeClass.a_type_wf_def NodeClass.get⇩N⇩o⇩d⇩e_type_wf ObjectClass.a_type_wf_def
bind.bind_lzero finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def l_type_wf_def⇩N⇩o⇩d⇩e.a_type_wf_def option.collapse
option.distinct(1) option.simps(3))
by (metis fmember.rep_eq)
lemma put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_shadow_root_opt_type_wf_preserved [simp]:
"h ⊢ put_M element_ptr shadow_root_opt_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply (metis NodeClass.a_type_wf_def NodeClass.get⇩N⇩o⇩d⇩e_type_wf ObjectClass.a_type_wf_def
bind.bind_lzero finite_set_in get⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def l_type_wf_def⇩N⇩o⇩d⇩e.a_type_wf_def option.collapse
option.distinct(1) option.simps(3))
by (metis fmember.rep_eq)
lemma new_character_data_type_wf_preserved [simp]:
"h ⊢ new_character_data →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: ElementMonad.put_M_defs put⇩E⇩l⇩e⇩m⇩e⇩n⇩t_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_kind_def
new_character_data_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def Let_def put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
dest!: get_heap_E
elim!: bind_returns_heap_E2 bind_returns_heap_E type_wf_put_ptr_not_in_heap_E
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
by (meson new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def new⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ptr_not_in_heap)
lemma new_character_data_is_l_new_character_data [instances]:
"l_new_character_data type_wf"
using l_new_character_data.intro new_character_data_type_wf_preserved
by blast
lemma put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_val_type_wf_preserved [simp]:
"h ⊢ put_M character_data_ptr val_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: CharacterDataMonad.put_M_defs put⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def put⇩N⇩o⇩d⇩e_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t is_node_kind_def
dest!: get_heap_E elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs CharacterDataMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply (metis bind.bind_lzero finite_set_in get⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_def option.distinct(1) option.exhaust_sel)
by (metis finite_set_in)
lemma new_document_type_wf_preserved [simp]: "h ⊢ new_document →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: new_document_def new⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def Let_def put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_ptr_kind_none
elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E
intro!: type_wf_put_I ElementMonad.type_wf_put_I CharacterDataMonad.type_wf_put_I
NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I
split: if_splits)[1]
apply(auto simp add: type_wf_defs ElementClass.type_wf_defs CharacterDataClass.type_wf_defs
NodeClass.type_wf_defs ObjectClass.type_wf_defs is_document_kind_def
split: option.splits)[1]
using document_ptrs_def apply fastforce
apply (simp add: is_document_kind_def)
apply (metis Suc_n_not_le_n document_ptr.sel(1) document_ptrs_def fMax_ge ffmember_filter
fimage_eqI is_document_ptr_ref)
done
locale l_new_document = l_type_wf +
assumes new_document_types_preserved: "h ⊢ new_document →⇩h h' ⟹ type_wf h = type_wf h'"
lemma new_document_is_l_new_document [instances]: "l_new_document type_wf"
using l_new_document.intro new_document_type_wf_preserved
by blast
lemma put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_doctype_type_wf_preserved [simp]:
"h ⊢ put_M document_ptr doctype_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: put_M_defs put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I
ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
apply(auto simp add: get_M_defs)[1]
by (metis (mono_tags) error_returns_result finite_set_in option.exhaust_sel option.simps(4))
lemma put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_document_element_type_wf_preserved [simp]:
"h ⊢ put_M document_ptr document_element_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: put_M_defs put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e
DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t is_node_ptr_kind_none
cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_none is_document_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I
ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: get_M_defs is_document_kind_def type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs
split: option.splits)[1]
by (metis finite_set_in)
lemma put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_disconnected_nodes_type_wf_preserved [simp]:
"h ⊢ put_M document_ptr disconnected_nodes_update v →⇩h h' ⟹ type_wf h = type_wf h'"
apply(auto simp add: put_M_defs put⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def
DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a
DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t
DocumentClass.type_wf⇩N⇩o⇩d⇩e
DocumentClass.type_wf⇩O⇩b⇩j⇩e⇩c⇩t
is_node_ptr_kind_none
cast⇩O⇩b⇩j⇩e⇩c⇩t⇩2⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_none is_document_kind_def
dest!: get_heap_E
elim!: bind_returns_heap_E2
intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I
ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I
ObjectMonad.type_wf_put_I)[1]
apply(auto simp add: is_document_kind_def get_M_defs type_wf_defs ElementClass.type_wf_defs
NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs
CharacterDataClass.type_wf_defs split: option.splits)[1]
by (metis finite_set_in)
lemma document_ptr_kinds_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "document_ptr_kinds h = document_ptr_kinds h'"
by(simp add: document_ptr_kinds_def preserved_def object_ptr_kinds_preserved_small[OF assms])
lemma document_ptr_kinds_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h'. ∀w ∈ SW. h ⊢ w →⇩h h'
⟶ (∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h')"
shows "document_ptr_kinds h = document_ptr_kinds h'"
using writes_small_big[OF assms]
apply(simp add: reflp_def transp_def preserved_def document_ptr_kinds_def)
by (metis assms object_ptr_kinds_preserved)
lemma type_wf_preserved_small:
assumes "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
assumes "⋀character_data_ptr. preserved
(get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr RCharacterData.nothing) h h'"
assumes "⋀document_ptr. preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr RDocument.nothing) h h'"
shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
using type_wf_preserved_small[OF assms(1) assms(2) assms(3) assms(4)]
allI[OF assms(5), of id, simplified] document_ptr_kinds_small[OF assms(1)]
apply(auto simp add: type_wf_defs )[1]
apply(auto simp add: type_wf_defs preserved_def get_M_defs document_ptr_kinds_small[OF assms(1)]
split: option.splits)[1]
apply force
apply(auto simp add: type_wf_defs preserved_def get_M_defs document_ptr_kinds_small[OF assms(1)]
split: option.splits)[1]
by force
lemma type_wf_preserved:
assumes "writes SW setter h h'"
assumes "h ⊢ setter →⇩h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀node_ptr. preserved (get_M⇩N⇩o⇩d⇩e node_ptr RNode.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀element_ptr. preserved (get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t element_ptr RElement.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀character_data_ptr. preserved
(get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a character_data_ptr RCharacterData.nothing) h h'"
assumes "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀document_ptr. preserved (get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t document_ptr RDocument.nothing) h h'"
shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
proof -
have "⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h' ⟹ DocumentClass.type_wf h = DocumentClass.type_wf h'"
using assms type_wf_preserved_small by fast
with assms(1) assms(2) show ?thesis
apply(rule writes_small_big)
by(auto simp add: reflp_def transp_def)
qed
lemma type_wf_drop: "type_wf h ⟹ type_wf (Heap (fmdrop ptr (the_heap h)))"
apply(auto simp add: type_wf_defs)[1]
using type_wf_drop
apply blast
by (metis (no_types, lifting) CharacterDataClass.get⇩O⇩b⇩j⇩e⇩c⇩t_type_wf CharacterDataMonad.type_wf_drop
document_ptr_kinds_commutes finite_set_in fmlookup_drop get⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_def get⇩O⇩b⇩j⇩e⇩c⇩t_def heap.sel)
end
Theory Core_DOM_Functions
section‹Querying and Modifying the DOM›
text‹In this theory, we are formalizing the functions for querying and modifying
the DOM.›
theory Core_DOM_Functions
imports
"monads/DocumentMonad"
begin
text ‹If we do not declare show\_variants, then all abbreviations that contain
constants that are overloaded by using adhoc\_overloading get immediately unfolded.›
declare [[show_variants]]
subsection ‹Various Functions›
lemma insort_split: "x ∈ set (insort y xs) ⟷ (x = y ∨ x ∈ set xs)"
apply(induct xs)
by(auto)
lemma concat_map_distinct:
"distinct (concat (map f xs)) ⟹ y ∈ set (concat (map f xs)) ⟹ ∃!x ∈ set xs. y ∈ set (f x)"
apply(induct xs)
by(auto)
lemma concat_map_all_distinct: "distinct (concat (map f xs)) ⟹ x ∈ set xs ⟹ distinct (f x)"
apply(induct xs)
by(auto)
lemma distinct_concat_map_I:
assumes "distinct xs"
and "⋀x. x ∈ set xs ⟹ distinct (f x)"
and "⋀x y. x ∈ set xs ⟹ y ∈ set xs ⟹ x ≠ y ⟹ (set (f x)) ∩ (set (f y)) = {}"
shows "distinct (concat ((map f xs)))"
using assms
apply(induct xs)
by(auto)
lemma distinct_concat_map_E:
assumes "distinct (concat ((map f xs)))"
shows "⋀x y. x ∈ set xs ⟹ y ∈ set xs ⟹ x ≠ y ⟹ (set (f x)) ∩ (set (f y)) = {}"
and "⋀x. x ∈ set xs ⟹ distinct (f x)"
using assms
apply(induct xs)
by(auto)
lemma bind_is_OK_E3 [elim]:
assumes "h ⊢ ok (f ⤜ g)" and "pure f h"
obtains x where "h ⊢ f →⇩r x" and "h ⊢ ok (g x)"
using assms
by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def pure_def
split: sum.splits)
subsection ‹Basic Functions›
subsubsection ‹get\_child\_nodes›
locale l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) element_ptr ⇒ unit ⇒ (_, (_) node_ptr list) dom_prog"
where
"get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r element_ptr _ = get_M element_ptr RElement.child_nodes"
definition get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r :: "(_) character_data_ptr ⇒ unit ⇒ (_, (_) node_ptr list) dom_prog"
where
"get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r _ _ = return []"
definition get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) document_ptr ⇒ unit ⇒ (_, (_) node_ptr list) dom_prog"
where
"get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr _ = do {
doc_elem ← get_M document_ptr document_element;
(case doc_elem of
Some element_ptr ⇒ return [cast element_ptr]
| None ⇒ return [])
}"
definition a_get_child_nodes_tups :: "(((_) object_ptr ⇒ bool) × ((_) object_ptr ⇒ unit
⇒ (_, (_) node_ptr list) dom_prog)) list"
where
"a_get_child_nodes_tups = [
(is_element_ptr, get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_character_data_ptr, get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_document_ptr, get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast)
]"
definition a_get_child_nodes :: "(_) object_ptr ⇒ (_, (_) node_ptr list) dom_prog"
where
"a_get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()"
definition a_get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
where
"a_get_child_nodes_locs ptr ≡
(if is_element_ptr_kind ptr then {preserved (get_M (the (cast ptr)) RElement.child_nodes)} else {}) ∪
(if is_document_ptr_kind ptr then {preserved (get_M (the (cast ptr)) RDocument.document_element)} else {}) ∪
{preserved (get_M ptr RObject.nothing)}"
definition first_child :: "(_) object_ptr ⇒ (_, (_) node_ptr option) dom_prog"
where
"first_child ptr = do {
children ← a_get_child_nodes ptr;
return (case children of [] ⇒ None | child#_ ⇒ Some child)}"
end
locale l_get_child_nodes_defs =
fixes get_child_nodes :: "(_) object_ptr ⇒ (_, (_) node_ptr list) dom_prog"
fixes get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_known_ptr known_ptr +
l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ (_, (_) node_ptr list) dom_prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set" +
assumes known_ptr_impl: "known_ptr = DocumentClass.known_ptr"
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes get_child_nodes_impl: "get_child_nodes = a_get_child_nodes"
assumes get_child_nodes_locs_impl: "get_child_nodes_locs = a_get_child_nodes_locs"
begin
lemmas get_child_nodes_def = get_child_nodes_impl[unfolded a_get_child_nodes_def]
lemmas get_child_nodes_locs_def = get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def]
lemma get_child_nodes_split:
"P (invoke (a_get_child_nodes_tups @ xs) ptr ()) =
((known_ptr ptr ⟶ P (get_child_nodes ptr))
∧ (¬(known_ptr ptr) ⟶ P (invoke xs ptr ())))"
by(auto simp add: known_ptr_impl get_child_nodes_impl a_get_child_nodes_def a_get_child_nodes_tups_def
known_ptr_defs CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs
NodeClass.known_ptr_defs
split: invoke_splits)
lemma get_child_nodes_split_asm:
"P (invoke (a_get_child_nodes_tups @ xs) ptr ()) =
(¬((known_ptr ptr ∧ ¬P (get_child_nodes ptr))
∨ (¬(known_ptr ptr) ∧ ¬P (invoke xs ptr ()))))"
by(auto simp add: known_ptr_impl get_child_nodes_impl a_get_child_nodes_def
a_get_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: invoke_splits)
lemmas get_child_nodes_splits = get_child_nodes_split get_child_nodes_split_asm
lemma get_child_nodes_ok [simp]:
assumes "known_ptr ptr"
assumes "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
shows "h ⊢ ok (get_child_nodes ptr)"
using assms(1) assms(2) assms(3)
apply(auto simp add: known_ptr_impl type_wf_impl get_child_nodes_def a_get_child_nodes_tups_def)[1]
apply(split invoke_splits, rule conjI)+
apply((rule impI)+, drule(1) known_ptr_not_document_ptr, drule(1) known_ptr_not_character_data_ptr,
drule(1) known_ptr_not_element_ptr)
apply(auto simp add: NodeClass.known_ptr_defs)[1]
apply(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def dest: get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok
split: list.splits option.splits intro!: bind_is_OK_I2)[1]
apply(auto simp add: get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def)[1]
apply (auto simp add: get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def CharacterDataClass.type_wf_defs
DocumentClass.type_wf_defs intro!: bind_is_OK_I2 split: option.splits)[1]
using get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok ‹type_wf h›[unfolded type_wf_impl] by blast
lemma get_child_nodes_ptr_in_heap [simp]:
assumes "h ⊢ get_child_nodes ptr →⇩r children"
shows "ptr |∈| object_ptr_kinds h"
using assms
by(auto simp add: get_child_nodes_impl a_get_child_nodes_def invoke_ptr_in_heap
dest: is_OK_returns_result_I)
lemma get_child_nodes_pure [simp]:
"pure (get_child_nodes ptr) h"
apply (auto simp add: get_child_nodes_impl a_get_child_nodes_def a_get_child_nodes_tups_def)[1]
apply(split invoke_splits, rule conjI)+
by(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro!: bind_pure_I split: option.splits)
lemma get_child_nodes_reads: "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"
apply(simp add: get_child_nodes_locs_impl get_child_nodes_impl a_get_child_nodes_def
a_get_child_nodes_tups_def a_get_child_nodes_locs_def)
apply(split invoke_splits, rule conjI)+
apply(auto)[1]
apply(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro: reads_subset[OF reads_singleton]
reads_subset[OF check_in_heap_reads]
intro!: reads_bind_pure reads_subset[OF return_reads] split: option.splits)[1]
apply(auto simp add: get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def intro: reads_subset[OF check_in_heap_reads]
intro!: reads_bind_pure reads_subset[OF return_reads] )[1]
apply(auto simp add: get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro: reads_subset[OF reads_singleton]
reads_subset[OF check_in_heap_reads] intro!: reads_bind_pure reads_subset[OF return_reads]
split: option.splits)[1]
done
end
locale l_get_child_nodes = l_type_wf + l_known_ptr + l_get_child_nodes_defs +
assumes get_child_nodes_reads: "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"
assumes get_child_nodes_ok: "type_wf h ⟹ known_ptr ptr ⟹ ptr |∈| object_ptr_kinds h
⟹ h ⊢ ok (get_child_nodes ptr)"
assumes get_child_nodes_ptr_in_heap: "h ⊢ ok (get_child_nodes ptr) ⟹ ptr |∈| object_ptr_kinds h"
assumes get_child_nodes_pure [simp]: "pure (get_child_nodes ptr) h"
global_interpretation l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
get_child_nodes = l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_child_nodes and
get_child_nodes_locs = l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_child_nodes_locs
.
interpretation
i_get_child_nodes?: l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
by(auto simp add: l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def get_child_nodes_def get_child_nodes_locs_def)
declare l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_child_nodes_is_l_get_child_nodes [instances]:
"l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
apply(unfold_locales)
using get_child_nodes_reads get_child_nodes_ok get_child_nodes_ptr_in_heap get_child_nodes_pure
by blast+
paragraph ‹new\_element›
locale l_new_element_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_child_nodes_new_element:
"ptr' ≠ cast new_element_ptr ⟹ h ⊢ new_element →⇩r new_element_ptr ⟹ h ⊢ new_element →⇩h h'
⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
by (auto simp add: get_child_nodes_locs_def new_element_get_M⇩O⇩b⇩j⇩e⇩c⇩t new_element_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t
new_element_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t split: prod.splits if_splits option.splits
elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)
lemma new_element_no_child_nodes:
"h ⊢ new_element →⇩r new_element_ptr ⟹ h ⊢ new_element →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_element_ptr) →⇩r []"
apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def
split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)[1]
apply(split invoke_splits, rule conjI)+
apply(auto intro: new_element_is_element_ptr)[1]
by(auto simp add: new_element_ptr_in_heap get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def check_in_heap_def
new_element_child_nodes intro!: bind_pure_returns_result_I
intro: new_element_is_element_ptr elim!: new_element_ptr_in_heap)
end
locale l_new_element_get_child_nodes = l_new_element + l_get_child_nodes +
assumes get_child_nodes_new_element:
"ptr' ≠ cast new_element_ptr ⟹ h ⊢ new_element →⇩r new_element_ptr
⟹ h ⊢ new_element →⇩h h' ⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
assumes new_element_no_child_nodes:
"h ⊢ new_element →⇩r new_element_ptr ⟹ h ⊢ new_element →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_element_ptr) →⇩r []"
interpretation i_new_element_get_child_nodes?:
l_new_element_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
by(unfold_locales)
declare l_new_element_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma new_element_get_child_nodes_is_l_new_element_get_child_nodes [instances]:
"l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
using new_element_is_l_new_element get_child_nodes_is_l_get_child_nodes
apply(auto simp add: l_new_element_get_child_nodes_def l_new_element_get_child_nodes_axioms_def)[1]
using get_child_nodes_new_element new_element_no_child_nodes
by fast+
paragraph ‹new\_character\_data›
locale l_new_character_data_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_child_nodes_new_character_data:
"ptr' ≠ cast new_character_data_ptr ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ h ⊢ new_character_data →⇩h h' ⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
by (auto simp add: get_child_nodes_locs_def new_character_data_get_M⇩O⇩b⇩j⇩e⇩c⇩t
new_character_data_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t new_character_data_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t
split: prod.splits if_splits option.splits
elim!: bind_returns_result_E bind_returns_heap_E
intro: is_character_data_ptr_kind_obtains)
lemma new_character_data_no_child_nodes:
"h ⊢ new_character_data →⇩r new_character_data_ptr ⟹ h ⊢ new_character_data →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []"
apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def
split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)[1]
apply(split invoke_splits, rule conjI)+
apply(auto intro: new_character_data_is_character_data_ptr)[1]
by(auto simp add: new_character_data_ptr_in_heap get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def
check_in_heap_def new_character_data_child_nodes
intro!: bind_pure_returns_result_I
intro: new_character_data_is_character_data_ptr elim!: new_character_data_ptr_in_heap)
end
locale l_new_character_data_get_child_nodes = l_new_character_data + l_get_child_nodes +
assumes get_child_nodes_new_character_data:
"ptr' ≠ cast new_character_data_ptr ⟹ h ⊢ new_character_data →⇩r new_character_data_ptr
⟹ h ⊢ new_character_data →⇩h h' ⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
assumes new_character_data_no_child_nodes:
"h ⊢ new_character_data →⇩r new_character_data_ptr ⟹ h ⊢ new_character_data →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []"
interpretation i_new_character_data_get_child_nodes?:
l_new_character_data_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
by(unfold_locales)
declare l_new_character_data_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma new_character_data_get_child_nodes_is_l_new_character_data_get_child_nodes [instances]:
"l_new_character_data_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
using new_character_data_is_l_new_character_data get_child_nodes_is_l_get_child_nodes
apply(simp add: l_new_character_data_get_child_nodes_def l_new_character_data_get_child_nodes_axioms_def)
using get_child_nodes_new_character_data new_character_data_no_child_nodes
by fast
paragraph ‹new\_document›
locale l_new_document_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_child_nodes_new_document:
"ptr' ≠ cast new_document_ptr ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ h ⊢ new_document →⇩h h' ⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
by (auto simp add: get_child_nodes_locs_def new_document_get_M⇩O⇩b⇩j⇩e⇩c⇩t new_document_get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t
new_document_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t split: prod.splits if_splits option.splits
elim!: bind_returns_result_E bind_returns_heap_E
intro: is_document_ptr_kind_obtains)
lemma new_document_no_child_nodes:
"h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []"
apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def
split: prod.splits
elim!: bind_returns_result_E bind_returns_heap_E)[1]
apply(split invoke_splits, rule conjI)+
apply(auto intro: new_document_is_document_ptr)[1]
by(auto simp add: new_document_ptr_in_heap get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def check_in_heap_def
new_document_document_element
intro!: bind_pure_returns_result_I
intro: new_document_is_document_ptr elim!: new_document_ptr_in_heap split: option.splits)
end
locale l_new_document_get_child_nodes = l_new_document + l_get_child_nodes +
assumes get_child_nodes_new_document:
"ptr' ≠ cast new_document_ptr ⟹ h ⊢ new_document →⇩r new_document_ptr
⟹ h ⊢ new_document →⇩h h' ⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
assumes new_document_no_child_nodes:
"h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []"
interpretation i_new_document_get_child_nodes?:
l_new_document_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr get_child_nodes get_child_nodes_locs
by(unfold_locales)
declare l_new_document_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma new_document_get_child_nodes_is_l_new_document_get_child_nodes [instances]:
"l_new_document_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
using new_document_is_l_new_document get_child_nodes_is_l_get_child_nodes
apply(simp add: l_new_document_get_child_nodes_def l_new_document_get_child_nodes_axioms_def)
using get_child_nodes_new_document new_document_no_child_nodes
by fast
subsubsection ‹set\_child\_nodes›
locale l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ::
"(_) element_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
where
"set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r element_ptr children = put_M element_ptr RElement.child_nodes_update children"
definition set_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ::
"(_) character_data_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
where
"set_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r _ _ = error HierarchyRequestError"
definition set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
where
"set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr children = do {
(case children of
[] ⇒ put_M document_ptr document_element_update None
| child # [] ⇒ (case cast child of
Some element_ptr ⇒ put_M document_ptr document_element_update (Some element_ptr)
| None ⇒ error HierarchyRequestError)
| _ ⇒ error HierarchyRequestError)
}"
definition a_set_child_nodes_tups ::
"(((_) object_ptr ⇒ bool) × ((_) object_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog)) list"
where
"a_set_child_nodes_tups ≡ [
(is_element_ptr, set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_character_data_ptr, set_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_document_ptr, set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast)
]"
definition a_set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
where
"a_set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr (children)"
lemmas set_child_nodes_defs = a_set_child_nodes_def
definition a_set_child_nodes_locs :: "(_) object_ptr ⇒ (_, unit) dom_prog set"
where
"a_set_child_nodes_locs ptr ≡
(if is_element_ptr_kind ptr
then all_args (put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t (the (cast ptr)) RElement.child_nodes_update) else {}) ∪
(if is_document_ptr_kind ptr
then all_args (put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t (the (cast ptr)) document_element_update) else {})"
end
locale l_set_child_nodes_defs =
fixes set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
fixes set_child_nodes_locs :: "(_) object_ptr ⇒ (_, unit) dom_prog set"
locale l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_known_ptr known_ptr +
l_set_child_nodes_defs set_child_nodes set_child_nodes_locs +
l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
and set_child_nodes_locs :: "(_) object_ptr ⇒ (_, unit) dom_prog set" +
assumes known_ptr_impl: "known_ptr = DocumentClass.known_ptr"
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes set_child_nodes_impl: "set_child_nodes = a_set_child_nodes"
assumes set_child_nodes_locs_impl: "set_child_nodes_locs = a_set_child_nodes_locs"
begin
lemmas set_child_nodes_def = set_child_nodes_impl[unfolded a_set_child_nodes_def]
lemmas set_child_nodes_locs_def = set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def]
lemma set_child_nodes_split:
"P (invoke (a_set_child_nodes_tups @ xs) ptr (children)) =
((known_ptr ptr ⟶ P (set_child_nodes ptr children))
∧ (¬(known_ptr ptr) ⟶ P (invoke xs ptr (children))))"
by(auto simp add: known_ptr_impl set_child_nodes_impl a_set_child_nodes_def
a_set_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: invoke_splits)
lemma set_child_nodes_split_asm:
"P (invoke (a_set_child_nodes_tups @ xs) ptr (children)) =
(¬((known_ptr ptr ∧ ¬P (set_child_nodes ptr children))
∨ (¬(known_ptr ptr) ∧ ¬P (invoke xs ptr (children)))))"
by(auto simp add: known_ptr_impl set_child_nodes_impl a_set_child_nodes_def
a_set_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: invoke_splits)[1]
lemmas set_child_nodes_splits = set_child_nodes_split set_child_nodes_split_asm
lemma set_child_nodes_writes: "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"
apply(simp add: set_child_nodes_locs_impl set_child_nodes_impl a_set_child_nodes_def
a_set_child_nodes_tups_def a_set_child_nodes_locs_def)
apply(split invoke_splits, rule conjI)+
apply(auto)[1]
apply(auto simp add: set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro!: writes_bind_pure
intro: writes_union_right_I split: list.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto intro: writes_union_right_I split: option.splits)[1]
apply(auto simp add: set_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def intro!: writes_bind_pure)[1]
apply(auto simp add: set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro: writes_union_left_I
intro!: writes_bind_pure split: list.splits option.splits)[1]
done
lemma set_child_nodes_pointers_preserved:
assumes "w ∈ set_child_nodes_locs object_ptr"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: set_child_nodes_locs_impl all_args_def a_set_child_nodes_locs_def
split: if_splits)
lemma set_child_nodes_typess_preserved:
assumes "w ∈ set_child_nodes_locs object_ptr"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: set_child_nodes_locs_impl type_wf_impl all_args_def a_set_child_nodes_locs_def
split: if_splits)
end
locale l_set_child_nodes = l_type_wf + l_set_child_nodes_defs +
assumes set_child_nodes_writes:
"writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"
assumes set_child_nodes_pointers_preserved:
"w ∈ set_child_nodes_locs object_ptr ⟹ h ⊢ w →⇩h h' ⟹ object_ptr_kinds h = object_ptr_kinds h'"
assumes set_child_nodes_types_preserved:
"w ∈ set_child_nodes_locs object_ptr ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
global_interpretation l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
set_child_nodes = l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_child_nodes and
set_child_nodes_locs = l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_child_nodes_locs .
interpretation
i_set_child_nodes?: l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr set_child_nodes set_child_nodes_locs
apply(unfold_locales)
by (auto simp add: set_child_nodes_def set_child_nodes_locs_def)
declare l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_child_nodes_is_l_set_child_nodes [instances]:
"l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs"
apply(unfold_locales)
using set_child_nodes_pointers_preserved set_child_nodes_typess_preserved set_child_nodes_writes
by blast+
paragraph ‹get\_child\_nodes›
locale l_set_child_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M = l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M + l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_child_nodes_get_child_nodes:
assumes "known_ptr ptr"
assumes "type_wf h"
assumes "h ⊢ set_child_nodes ptr children →⇩h h'"
shows "h' ⊢ get_child_nodes ptr →⇩r children"
proof -
have "h ⊢ check_in_heap ptr →⇩r ()"
using assms set_child_nodes_impl[unfolded a_set_child_nodes_def] invoke_ptr_in_heap
by (metis (full_types) check_in_heap_ptr_in_heap is_OK_returns_heap_I is_OK_returns_result_E
old.unit.exhaust)
then have ptr_in_h: "ptr |∈| object_ptr_kinds h"
by (simp add: check_in_heap_ptr_in_heap is_OK_returns_result_I)
have "type_wf h'"
apply(unfold type_wf_impl)
apply(rule subst[where P=id, OF type_wf_preserved[OF set_child_nodes_writes assms(3),
unfolded all_args_def], simplified])
by(auto simp add: all_args_def assms(2)[unfolded type_wf_impl]
set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def]
split: if_splits)
have "h' ⊢ check_in_heap ptr →⇩r ()"
using check_in_heap_reads set_child_nodes_writes assms(3) ‹h ⊢ check_in_heap ptr →⇩r ()›
apply(rule reads_writes_separate_forwards)
by(auto simp add: all_args_def set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def])
then have "ptr |∈| object_ptr_kinds h'"
using check_in_heap_ptr_in_heap by blast
with assms ptr_in_h ‹type_wf h'› show ?thesis
apply(auto simp add: get_child_nodes_impl set_child_nodes_impl type_wf_impl known_ptr_impl
a_get_child_nodes_def a_get_child_nodes_tups_def a_set_child_nodes_def
a_set_child_nodes_tups_def
del: bind_pure_returns_result_I2
intro!: bind_pure_returns_result_I2)[1]
apply(split invoke_splits, rule conjI)
apply(split invoke_splits, rule conjI)
apply(split invoke_splits, rule conjI)
apply(auto simp add: NodeClass.known_ptr_defs
dest!: known_ptr_not_document_ptr known_ptr_not_character_data_ptr
known_ptr_not_element_ptr)[1]
apply(auto simp add: NodeClass.known_ptr_defs
dest!: known_ptr_not_document_ptr known_ptr_not_character_data_ptr
known_ptr_not_element_ptr)[1]
apply(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok
split: list.splits option.splits
intro!: bind_pure_returns_result_I2
dest: get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok; auto dest: returns_result_eq
dest!: document_put_get[where getter = document_element])[1]
apply(auto simp add: get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def set_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def)[1]
by(auto simp add: get_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def dest: element_put_get)
qed
lemma set_child_nodes_get_child_nodes_different_pointers:
assumes "ptr ≠ ptr'"
assumes "w ∈ set_child_nodes_locs ptr"
assumes "h ⊢ w →⇩h h'"
assumes "r ∈ get_child_nodes_locs ptr'"
shows "r h h'"
using assms
apply(auto simp add: get_child_nodes_locs_impl set_child_nodes_locs_impl all_args_def
a_set_child_nodes_locs_def a_get_child_nodes_locs_def
split: if_splits option.splits )[1]
apply(rule is_document_ptr_kind_obtains)
apply(simp)
apply(rule is_document_ptr_kind_obtains)
apply(auto)[1]
apply(auto)[1]
apply(rule is_element_ptr_kind_obtains)
apply(auto)[1]
apply(auto)[1]
apply(rule is_element_ptr_kind_obtains)
apply(auto)[1]
apply(auto)[1]
done
lemma set_child_nodes_element_ok [simp]:
assumes "known_ptr ptr"
assumes "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
assumes "is_element_ptr_kind ptr"
shows "h ⊢ ok (set_child_nodes ptr children)"
proof -
have "is_element_ptr ptr"
using ‹known_ptr ptr› assms(4)
by(auto simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then show ?thesis
using assms
apply(auto simp add: set_child_nodes_def a_set_child_nodes_tups_def set_child_nodes⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
split: option.splits)[1]
by (simp add: DocumentMonad.put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok local.type_wf_impl)
qed
lemma set_child_nodes_document1_ok [simp]:
assumes "known_ptr ptr"
assumes "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
assumes "is_document_ptr_kind ptr"
assumes "children = []"
shows "h ⊢ ok (set_child_nodes ptr children)"
proof -
have "is_document_ptr ptr"
using ‹known_ptr ptr› assms(4)
by(auto simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then show ?thesis
using assms
apply(auto simp add: set_child_nodes_def a_set_child_nodes_tups_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
split: option.splits)[1]
by (simp add: DocumentMonad.put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok local.type_wf_impl)
qed
lemma set_child_nodes_document2_ok [simp]:
assumes "known_ptr ptr"
assumes "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
assumes "is_document_ptr_kind ptr"
assumes "children = [child]"
assumes "is_element_ptr_kind child"
shows "h ⊢ ok (set_child_nodes ptr children)"
proof -
have "is_document_ptr ptr"
using ‹known_ptr ptr› assms(4)
by(auto simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then show ?thesis
using assms
apply(auto simp add: set_child_nodes_def a_set_child_nodes_tups_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def)[1]
apply(split invoke_splits, rule conjI)+
apply(auto simp add: is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def split: option.splits)[1]
apply(auto simp add: is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def split: option.splits)[1]
apply (simp add: local.type_wf_impl put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok)
apply(auto simp add: is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def split: option.splits)[1]
by(auto simp add: is_element_ptr_kind⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def set_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def split: option.splits)[1]
qed
end
locale l_set_child_nodes_get_child_nodes = l_get_child_nodes + l_set_child_nodes +
assumes set_child_nodes_get_child_nodes:
"type_wf h ⟹ known_ptr ptr
⟹ h ⊢ set_child_nodes ptr children →⇩h h' ⟹ h' ⊢ get_child_nodes ptr →⇩r children"
assumes set_child_nodes_get_child_nodes_different_pointers:
"ptr ≠ ptr' ⟹ w ∈ set_child_nodes_locs ptr ⟹ h ⊢ w →⇩h h'
⟹ r ∈ get_child_nodes_locs ptr' ⟹ r h h'"
interpretation
i_set_child_nodes_get_child_nodes?: l_set_child_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs
by unfold_locales
declare l_set_child_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_child_nodes_get_child_nodes_is_l_set_child_nodes_get_child_nodes [instances]:
"l_set_child_nodes_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs
set_child_nodes set_child_nodes_locs"
using get_child_nodes_is_l_get_child_nodes set_child_nodes_is_l_set_child_nodes
apply(auto simp add: l_set_child_nodes_get_child_nodes_def l_set_child_nodes_get_child_nodes_axioms_def)[1]
using set_child_nodes_get_child_nodes apply blast
using set_child_nodes_get_child_nodes_different_pointers apply metis
done
subsubsection ‹get\_attribute›
locale l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_get_attribute :: "(_) element_ptr ⇒ attr_key ⇒ (_, attr_value option) dom_prog"
where
"a_get_attribute ptr k = do {m ← get_M ptr attrs; return (fmlookup m k)}"
lemmas get_attribute_defs = a_get_attribute_def
definition a_get_attribute_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
where
"a_get_attribute_locs element_ptr = {preserved (get_M element_ptr attrs)}"
end
locale l_get_attribute_defs =
fixes get_attribute :: "(_) element_ptr ⇒ attr_key ⇒ (_, attr_value option) dom_prog"
fixes get_attribute_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_get_attribute_defs get_attribute get_attribute_locs +
l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and get_attribute :: "(_) element_ptr ⇒ attr_key ⇒ (_, attr_value option) dom_prog"
and get_attribute_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes get_attribute_impl: "get_attribute = a_get_attribute"
assumes get_attribute_locs_impl: "get_attribute_locs = a_get_attribute_locs"
begin
lemma get_attribute_pure [simp]: "pure (get_attribute ptr k) h"
by (auto simp add: bind_pure_I get_attribute_impl[unfolded a_get_attribute_def])
lemma get_attribute_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (get_attribute element_ptr k)"
apply(unfold type_wf_impl)
unfolding get_attribute_impl[unfolded a_get_attribute_def] using get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok
by (metis bind_is_OK_pure_I return_ok ElementMonad.get_M_pure)
lemma get_attribute_ptr_in_heap:
"h ⊢ ok (get_attribute element_ptr k) ⟹ element_ptr |∈| element_ptr_kinds h"
unfolding get_attribute_impl[unfolded a_get_attribute_def]
by (meson DocumentMonad.get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap bind_is_OK_E is_OK_returns_result_I)
lemma get_attribute_reads:
"reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h'"
by(auto simp add: get_attribute_impl[unfolded a_get_attribute_def]
get_attribute_locs_impl[unfolded a_get_attribute_locs_def]
reads_insert_writes_set_right
intro!: reads_bind_pure)
end
locale l_get_attribute = l_type_wf + l_get_attribute_defs +
assumes get_attribute_reads:
"reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h'"
assumes get_attribute_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (get_attribute element_ptr k)"
assumes get_attribute_ptr_in_heap:
"h ⊢ ok (get_attribute element_ptr k) ⟹ element_ptr |∈| element_ptr_kinds h"
assumes get_attribute_pure [simp]: "pure (get_attribute element_ptr k) h"
global_interpretation l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
get_attribute = l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_attribute and
get_attribute_locs = l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_attribute_locs .
interpretation
i_get_attribute?: l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_attribute get_attribute_locs
apply(unfold_locales)
by (auto simp add: get_attribute_def get_attribute_locs_def)
declare l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_attribute_is_l_get_attribute [instances]:
"l_get_attribute type_wf get_attribute get_attribute_locs"
apply(unfold_locales)
using get_attribute_reads get_attribute_ok get_attribute_ptr_in_heap get_attribute_pure
by blast+
subsubsection ‹set\_attribute›
locale l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition
a_set_attribute :: "(_) element_ptr ⇒ attr_key ⇒ attr_value option ⇒ (_, unit) dom_prog"
where
"a_set_attribute ptr k v = do {
m ← get_M ptr attrs;
put_M ptr attrs_update (if v = None then fmdrop k m else fmupd k (the v) m)
}"
definition a_set_attribute_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set"
where
"a_set_attribute_locs element_ptr ≡ all_args (put_M element_ptr attrs_update)"
end
locale l_set_attribute_defs =
fixes set_attribute :: "(_) element_ptr ⇒ attr_key ⇒ attr_value option ⇒ (_, unit) dom_prog"
fixes set_attribute_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set"
locale l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_set_attribute_defs set_attribute set_attribute_locs +
l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and set_attribute :: "(_) element_ptr ⇒ attr_key ⇒ attr_value option ⇒ (_, unit) dom_prog"
and set_attribute_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes set_attribute_impl: "set_attribute = a_set_attribute"
assumes set_attribute_locs_impl: "set_attribute_locs = a_set_attribute_locs"
begin
lemmas set_attribute_def = set_attribute_impl[folded a_set_attribute_def]
lemmas set_attribute_locs_def = set_attribute_locs_impl[unfolded a_set_attribute_locs_def]
lemma set_attribute_ok: "type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (set_attribute element_ptr k v)"
apply(unfold type_wf_impl)
unfolding set_attribute_impl[unfolded a_set_attribute_def] using get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok
by(metis (no_types, lifting) DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t ElementMonad.get_M_pure bind_is_OK_E
bind_is_OK_pure_I is_OK_returns_result_I)
lemma set_attribute_writes:
"writes (set_attribute_locs element_ptr) (set_attribute element_ptr k v) h h'"
by(auto simp add: set_attribute_impl[unfolded a_set_attribute_def]
set_attribute_locs_impl[unfolded a_set_attribute_locs_def]
intro: writes_bind_pure)
end
locale l_set_attribute = l_type_wf + l_set_attribute_defs +
assumes set_attribute_writes:
"writes (set_attribute_locs element_ptr) (set_attribute element_ptr k v) h h'"
assumes set_attribute_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (set_attribute element_ptr k v)"
global_interpretation l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
set_attribute = l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_attribute and
set_attribute_locs = l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_attribute_locs .
interpretation
i_set_attribute?: l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_attribute set_attribute_locs
apply(unfold_locales)
by (auto simp add: set_attribute_def set_attribute_locs_def)
declare l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_attribute_is_l_set_attribute [instances]:
"l_set_attribute type_wf set_attribute set_attribute_locs"
apply(unfold_locales)
using set_attribute_ok set_attribute_writes
by blast+
paragraph ‹get\_attribute›
locale l_set_attribute_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_attribute_get_attribute:
"h ⊢ set_attribute ptr k v →⇩h h' ⟹ h' ⊢ get_attribute ptr k →⇩r v"
by(auto simp add: set_attribute_impl[unfolded a_set_attribute_def]
get_attribute_impl[unfolded a_get_attribute_def]
elim!: bind_returns_heap_E2
intro!: bind_pure_returns_result_I
elim: element_put_get)
end
locale l_set_attribute_get_attribute = l_get_attribute + l_set_attribute +
assumes set_attribute_get_attribute:
"h ⊢ set_attribute ptr k v →⇩h h' ⟹ h' ⊢ get_attribute ptr k →⇩r v"
interpretation
i_set_attribute_get_attribute?: l_set_attribute_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
get_attribute get_attribute_locs set_attribute set_attribute_locs
by(unfold_locales)
declare l_set_attribute_get_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_attribute_get_attribute_is_l_set_attribute_get_attribute [instances]:
"l_set_attribute_get_attribute type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs"
using get_attribute_is_l_get_attribute set_attribute_is_l_set_attribute
apply(simp add: l_set_attribute_get_attribute_def l_set_attribute_get_attribute_axioms_def)
using set_attribute_get_attribute
by blast
paragraph ‹get\_child\_nodes›
locale l_set_attribute_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_attribute_get_child_nodes:
"∀w ∈ set_attribute_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
by(auto simp add: set_attribute_locs_def get_child_nodes_locs_def all_args_def
intro: element_put_get_preserved[where setter=attrs_update])
end
locale l_set_attribute_get_child_nodes =
l_set_attribute +
l_get_child_nodes +
assumes set_attribute_get_child_nodes:
"∀w ∈ set_attribute_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
interpretation
i_set_attribute_get_child_nodes?: l_set_attribute_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs
by unfold_locales
declare l_set_attribute_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_attribute_get_child_nodes_is_l_set_attribute_get_child_nodes [instances]:
"l_set_attribute_get_child_nodes type_wf set_attribute set_attribute_locs known_ptr
get_child_nodes get_child_nodes_locs"
using set_attribute_is_l_set_attribute get_child_nodes_is_l_get_child_nodes
apply(simp add: l_set_attribute_get_child_nodes_def l_set_attribute_get_child_nodes_axioms_def)
using set_attribute_get_child_nodes
by blast
subsubsection ‹get\_disconnected\_nodes›
locale l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_get_disconnected_nodes :: "(_) document_ptr
⇒ (_, (_) node_ptr list) dom_prog"
where
"a_get_disconnected_nodes document_ptr = get_M document_ptr disconnected_nodes"
lemmas get_disconnected_nodes_defs = a_get_disconnected_nodes_def
definition a_get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
where
"a_get_disconnected_nodes_locs document_ptr = {preserved (get_M document_ptr disconnected_nodes)}"
end
locale l_get_disconnected_nodes_defs =
fixes get_disconnected_nodes :: "(_) document_ptr ⇒ (_, (_) node_ptr list) dom_prog"
fixes get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes get_disconnected_nodes_impl: "get_disconnected_nodes = a_get_disconnected_nodes"
assumes get_disconnected_nodes_locs_impl: "get_disconnected_nodes_locs = a_get_disconnected_nodes_locs"
begin
lemmas
get_disconnected_nodes_def = get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def]
lemmas
get_disconnected_nodes_locs_def = get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def]
lemma get_disconnected_nodes_ok:
"type_wf h ⟹ document_ptr |∈| document_ptr_kinds h ⟹ h ⊢ ok (get_disconnected_nodes document_ptr)"
apply(unfold type_wf_impl)
unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] using get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok
by fast
lemma get_disconnected_nodes_ptr_in_heap:
"h ⊢ ok (get_disconnected_nodes document_ptr) ⟹ document_ptr |∈| document_ptr_kinds h"
unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def]
by (simp add: DocumentMonad.get_M_ptr_in_heap)
lemma get_disconnected_nodes_pure [simp]: "pure (get_disconnected_nodes document_ptr) h"
unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] by simp
lemma get_disconnected_nodes_reads:
"reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'"
by(simp add: get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def]
get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def]
reads_bind_pure reads_insert_writes_set_right)
end
locale l_get_disconnected_nodes = l_type_wf + l_get_disconnected_nodes_defs +
assumes get_disconnected_nodes_reads:
"reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'"
assumes get_disconnected_nodes_ok:
"type_wf h ⟹ document_ptr |∈| document_ptr_kinds h ⟹ h ⊢ ok (get_disconnected_nodes document_ptr)"
assumes get_disconnected_nodes_ptr_in_heap:
"h ⊢ ok (get_disconnected_nodes document_ptr) ⟹ document_ptr |∈| document_ptr_kinds h"
assumes get_disconnected_nodes_pure [simp]:
"pure (get_disconnected_nodes document_ptr) h"
global_interpretation l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
get_disconnected_nodes = l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_disconnected_nodes and
get_disconnected_nodes_locs = l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_disconnected_nodes_locs .
interpretation
i_get_disconnected_nodes?: l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes
get_disconnected_nodes_locs
apply(unfold_locales)
by (auto simp add: get_disconnected_nodes_def get_disconnected_nodes_locs_def)
declare l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_disconnected_nodes_is_l_get_disconnected_nodes [instances]:
"l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"
apply(simp add: l_get_disconnected_nodes_def)
using get_disconnected_nodes_reads get_disconnected_nodes_ok get_disconnected_nodes_ptr_in_heap
get_disconnected_nodes_pure
by blast+
paragraph ‹set\_child\_nodes›
locale l_set_child_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
CD: l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_child_nodes_get_disconnected_nodes:
"∀w ∈ a_set_child_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ a_get_disconnected_nodes_locs ptr'. r h h'))"
by(auto simp add: a_set_child_nodes_locs_def a_get_disconnected_nodes_locs_def all_args_def)
end
locale l_set_child_nodes_get_disconnected_nodes = l_set_child_nodes + l_get_disconnected_nodes +
assumes set_child_nodes_get_disconnected_nodes:
"∀w ∈ set_child_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
interpretation
i_set_child_nodes_get_disconnected_nodes?: l_set_child_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
known_ptr set_child_nodes set_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
by(unfold_locales)
declare l_set_child_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_child_nodes_get_disconnected_nodes_is_l_set_child_nodes_get_disconnected_nodes [instances]:
"l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs"
using set_child_nodes_is_l_set_child_nodes get_disconnected_nodes_is_l_get_disconnected_nodes
apply(simp add: l_set_child_nodes_get_disconnected_nodes_def
l_set_child_nodes_get_disconnected_nodes_axioms_def)
using set_child_nodes_get_disconnected_nodes
by fast
paragraph ‹set\_attribute›
locale l_set_attribute_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_attribute_get_disconnected_nodes:
"∀w ∈ a_set_attribute_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ a_get_disconnected_nodes_locs ptr'. r h h'))"
by(auto simp add: a_set_attribute_locs_def a_get_disconnected_nodes_locs_def all_args_def)
end
locale l_set_attribute_get_disconnected_nodes = l_set_attribute + l_get_disconnected_nodes +
assumes set_attribute_get_disconnected_nodes:
"∀w ∈ set_attribute_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
interpretation
i_set_attribute_get_disconnected_nodes?: l_set_attribute_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
set_attribute set_attribute_locs get_disconnected_nodes get_disconnected_nodes_locs
by(unfold_locales)
declare l_set_attribute_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_attribute_get_disconnected_nodes_is_l_set_attribute_get_disconnected_nodes [instances]:
"l_set_attribute_get_disconnected_nodes type_wf set_attribute set_attribute_locs
get_disconnected_nodes get_disconnected_nodes_locs"
using set_attribute_is_l_set_attribute get_disconnected_nodes_is_l_get_disconnected_nodes
apply(simp add: l_set_attribute_get_disconnected_nodes_def
l_set_attribute_get_disconnected_nodes_axioms_def)
using set_attribute_get_disconnected_nodes
by fast
paragraph ‹new\_element›
locale l_new_element_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_disconnected_nodes_new_element:
"h ⊢ new_element →⇩r new_element_ptr ⟹ h ⊢ new_element →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
by(auto simp add: get_disconnected_nodes_locs_def new_element_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t)
end
locale l_new_element_get_disconnected_nodes = l_get_disconnected_nodes_defs +
assumes get_disconnected_nodes_new_element:
"h ⊢ new_element →⇩r new_element_ptr ⟹ h ⊢ new_element →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
interpretation i_new_element_get_disconnected_nodes?:
l_new_element_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes
get_disconnected_nodes_locs
by unfold_locales
declare l_new_element_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma new_element_get_disconnected_nodes_is_l_new_element_get_disconnected_nodes [instances]:
"l_new_element_get_disconnected_nodes get_disconnected_nodes_locs"
by (simp add: get_disconnected_nodes_new_element l_new_element_get_disconnected_nodes_def)
paragraph ‹new\_character\_data›
locale l_new_character_data_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_disconnected_nodes_new_character_data:
"h ⊢ new_character_data →⇩r new_character_data_ptr ⟹ h ⊢ new_character_data →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
by(auto simp add: get_disconnected_nodes_locs_def new_character_data_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t)
end
locale l_new_character_data_get_disconnected_nodes = l_get_disconnected_nodes_defs +
assumes get_disconnected_nodes_new_character_data:
"h ⊢ new_character_data →⇩r new_character_data_ptr ⟹ h ⊢ new_character_data →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
interpretation i_new_character_data_get_disconnected_nodes?:
l_new_character_data_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes
get_disconnected_nodes_locs
by unfold_locales
declare l_new_character_data_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma new_character_data_get_disconnected_nodes_is_l_new_character_data_get_disconnected_nodes [instances]:
"l_new_character_data_get_disconnected_nodes get_disconnected_nodes_locs"
by (simp add: get_disconnected_nodes_new_character_data l_new_character_data_get_disconnected_nodes_def)
paragraph ‹new\_document›
locale l_new_document_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs
for type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_disconnected_nodes_new_document_different_pointers:
"new_document_ptr ≠ ptr' ⟹ h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
by(auto simp add: get_disconnected_nodes_locs_def new_document_get_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t)
lemma new_document_no_disconnected_nodes:
"h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []"
by(simp add: get_disconnected_nodes_def new_document_disconnected_nodes)
end
interpretation i_new_document_get_disconnected_nodes?:
l_new_document_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs
by unfold_locales
declare l_new_document_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
locale l_new_document_get_disconnected_nodes = l_get_disconnected_nodes_defs +
assumes get_disconnected_nodes_new_document_different_pointers:
"new_document_ptr ≠ ptr' ⟹ h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
assumes new_document_no_disconnected_nodes:
"h ⊢ new_document →⇩r new_document_ptr ⟹ h ⊢ new_document →⇩h h'
⟹ h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []"
lemma new_document_get_disconnected_nodes_is_l_new_document_get_disconnected_nodes [instances]:
"l_new_document_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs"
apply (auto simp add: l_new_document_get_disconnected_nodes_def)[1]
using get_disconnected_nodes_new_document_different_pointers apply fast
using new_document_no_disconnected_nodes apply blast
done
subsubsection ‹set\_disconnected\_nodes›
locale l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
where
"a_set_disconnected_nodes document_ptr disc_nodes =
put_M document_ptr disconnected_nodes_update disc_nodes"
lemmas set_disconnected_nodes_defs = a_set_disconnected_nodes_def
definition a_set_disconnected_nodes_locs :: "(_) document_ptr ⇒ (_, unit) dom_prog set"
where
"a_set_disconnected_nodes_locs document_ptr ≡ all_args (put_M document_ptr disconnected_nodes_update)"
end
locale l_set_disconnected_nodes_defs =
fixes set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
fixes set_disconnected_nodes_locs :: "(_) document_ptr ⇒ (_, unit) dom_prog set"
locale l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs +
l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ (_, unit) dom_prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ (_, unit) dom_prog set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes set_disconnected_nodes_impl: "set_disconnected_nodes = a_set_disconnected_nodes"
assumes set_disconnected_nodes_locs_impl: "set_disconnected_nodes_locs = a_set_disconnected_nodes_locs"
begin
lemmas set_disconnected_nodes_def = set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def]
lemmas set_disconnected_nodes_locs_def =
set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def]
lemma set_disconnected_nodes_ok:
"type_wf h ⟹ document_ptr |∈| document_ptr_kinds h ⟹
h ⊢ ok (set_disconnected_nodes document_ptr node_ptrs)"
by (simp add: type_wf_impl put_M⇩D⇩o⇩c⇩u⇩m⇩e⇩n⇩t_ok
set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def])
lemma set_disconnected_nodes_ptr_in_heap:
"h ⊢ ok (set_disconnected_nodes document_ptr disc_nodes) ⟹ document_ptr |∈| document_ptr_kinds h"
by (simp add: set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def]
DocumentMonad.put_M_ptr_in_heap)
lemma set_disconnected_nodes_writes:
"writes (set_disconnected_nodes_locs document_ptr) (set_disconnected_nodes document_ptr disc_nodes) h h'"
by(auto simp add: set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def]
set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def]
intro: writes_bind_pure)
lemma set_disconnected_nodes_pointers_preserved:
assumes "w ∈ set_disconnected_nodes_locs object_ptr"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: all_args_def set_disconnected_nodes_locs_impl[unfolded
a_set_disconnected_nodes_locs_def]
split: if_splits)
lemma set_disconnected_nodes_typess_preserved:
assumes "w ∈ set_disconnected_nodes_locs object_ptr"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
apply(unfold type_wf_impl)
by(auto simp add: all_args_def
set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def]
split: if_splits)
end
locale l_set_disconnected_nodes = l_type_wf + l_set_disconnected_nodes_defs +
assumes set_disconnected_nodes_writes:
"writes (set_disconnected_nodes_locs document_ptr)
(set_disconnected_nodes document_ptr disc_nodes) h h'"
assumes set_disconnected_nodes_ok:
"type_wf h ⟹ document_ptr |∈| document_ptr_kinds h ⟹
h ⊢ ok (set_disconnected_nodes document_ptr disc_noded)"
assumes set_disconnected_nodes_ptr_in_heap:
"h ⊢ ok (set_disconnected_nodes document_ptr disc_noded) ⟹
document_ptr |∈| document_ptr_kinds h"
assumes set_disconnected_nodes_pointers_preserved:
"w ∈ set_disconnected_nodes_locs document_ptr ⟹ h ⊢ w →⇩h h' ⟹
object_ptr_kinds h = object_ptr_kinds h'"
assumes set_disconnected_nodes_types_preserved:
"w ∈ set_disconnected_nodes_locs document_ptr ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
global_interpretation l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
set_disconnected_nodes = l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_disconnected_nodes and
set_disconnected_nodes_locs = l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_disconnected_nodes_locs .
interpretation
i_set_disconnected_nodes?: l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_disconnected_nodes
set_disconnected_nodes_locs
apply unfold_locales
by (auto simp add: set_disconnected_nodes_def set_disconnected_nodes_locs_def)
declare l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_is_l_set_disconnected_nodes [instances]:
"l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs"
apply(simp add: l_set_disconnected_nodes_def)
using set_disconnected_nodes_ok set_disconnected_nodes_writes
set_disconnected_nodes_pointers_preserved
set_disconnected_nodes_ptr_in_heap set_disconnected_nodes_typess_preserved
by blast+
paragraph ‹get\_disconnected\_nodes›
locale l_set_disconnected_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M = l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
+ l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_disconnected_nodes_get_disconnected_nodes:
assumes "h ⊢ a_set_disconnected_nodes document_ptr disc_nodes →⇩h h'"
shows "h' ⊢ a_get_disconnected_nodes document_ptr →⇩r disc_nodes"
using assms
by(auto simp add: a_get_disconnected_nodes_def a_set_disconnected_nodes_def)
lemma set_disconnected_nodes_get_disconnected_nodes_different_pointers:
assumes "ptr ≠ ptr'"
assumes "w ∈ a_set_disconnected_nodes_locs ptr"
assumes "h ⊢ w →⇩h h'"
assumes "r ∈ a_get_disconnected_nodes_locs ptr'"
shows "r h h'"
using assms
by(auto simp add: all_args_def a_set_disconnected_nodes_locs_def a_get_disconnected_nodes_locs_def
split: if_splits option.splits )
end
locale l_set_disconnected_nodes_get_disconnected_nodes = l_get_disconnected_nodes
+ l_set_disconnected_nodes +
assumes set_disconnected_nodes_get_disconnected_nodes:
"h ⊢ set_disconnected_nodes document_ptr disc_nodes →⇩h h'
⟹ h' ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assumes set_disconnected_nodes_get_disconnected_nodes_different_pointers:
"ptr ≠ ptr' ⟹ w ∈ set_disconnected_nodes_locs ptr ⟹ h ⊢ w →⇩h h'
⟹ r ∈ get_disconnected_nodes_locs ptr' ⟹ r h h'"
interpretation i_set_disconnected_nodes_get_disconnected_nodes?:
l_set_disconnected_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs
by unfold_locales
declare l_set_disconnected_nodes_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_disconnected_nodes_is_l_set_disconnected_nodes_get_disconnected_nodes
[instances]:
"l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs"
using set_disconnected_nodes_is_l_set_disconnected_nodes get_disconnected_nodes_is_l_get_disconnected_nodes
apply(simp add: l_set_disconnected_nodes_get_disconnected_nodes_def
l_set_disconnected_nodes_get_disconnected_nodes_axioms_def)
using set_disconnected_nodes_get_disconnected_nodes
set_disconnected_nodes_get_disconnected_nodes_different_pointers
by fast+
paragraph ‹get\_child\_nodes›
locale l_set_disconnected_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_disconnected_nodes_get_child_nodes:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
by(auto simp add: set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def]
get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def)
end
locale l_set_disconnected_nodes_get_child_nodes = l_set_disconnected_nodes_defs + l_get_child_nodes_defs +
assumes set_disconnected_nodes_get_child_nodes [simp]:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
interpretation
i_set_disconnected_nodes_get_child_nodes?: l_set_disconnected_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf
set_disconnected_nodes set_disconnected_nodes_locs
known_ptr get_child_nodes get_child_nodes_locs
by unfold_locales
declare l_set_disconnected_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_child_nodes_is_l_set_disconnected_nodes_get_child_nodes [instances]:
"l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
using set_disconnected_nodes_is_l_set_disconnected_nodes get_child_nodes_is_l_get_child_nodes
apply(simp add: l_set_disconnected_nodes_get_child_nodes_def)
using set_disconnected_nodes_get_child_nodes
by fast
subsubsection ‹get\_tag\_name›
locale l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_get_tag_name :: "(_) element_ptr ⇒ (_, tag_name) dom_prog"
where
"a_get_tag_name element_ptr = get_M element_ptr tag_name"
definition a_get_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
where
"a_get_tag_name_locs element_ptr ≡ {preserved (get_M element_ptr tag_name)}"
end
locale l_get_tag_name_defs =
fixes get_tag_name :: "(_) element_ptr ⇒ (_, tag_name) dom_prog"
fixes get_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_get_tag_name_defs get_tag_name get_tag_name_locs +
l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and get_tag_name :: "(_) element_ptr ⇒ (_, tag_name) dom_prog"
and get_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes get_tag_name_impl: "get_tag_name = a_get_tag_name"
assumes get_tag_name_locs_impl: "get_tag_name_locs = a_get_tag_name_locs"
begin
lemmas get_tag_name_def = get_tag_name_impl[unfolded a_get_tag_name_def]
lemmas get_tag_name_locs_def = get_tag_name_locs_impl[unfolded a_get_tag_name_locs_def]
lemma get_tag_name_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (get_tag_name element_ptr)"
apply(unfold type_wf_impl get_tag_name_impl[unfolded a_get_tag_name_def])
using get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok
by blast
lemma get_tag_name_pure [simp]: "pure (get_tag_name element_ptr) h"
unfolding get_tag_name_impl[unfolded a_get_tag_name_def]
by simp
lemma get_tag_name_ptr_in_heap [simp]:
assumes "h ⊢ get_tag_name element_ptr →⇩r children"
shows "element_ptr |∈| element_ptr_kinds h"
using assms
by(auto simp add: get_tag_name_impl[unfolded a_get_tag_name_def] get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ptr_in_heap
dest: is_OK_returns_result_I)
lemma get_tag_name_reads: "reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h'"
by(simp add: get_tag_name_impl[unfolded a_get_tag_name_def]
get_tag_name_locs_impl[unfolded a_get_tag_name_locs_def] reads_bind_pure
reads_insert_writes_set_right)
end
locale l_get_tag_name = l_type_wf + l_get_tag_name_defs +
assumes get_tag_name_reads:
"reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h'"
assumes get_tag_name_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (get_tag_name element_ptr)"
assumes get_tag_name_ptr_in_heap:
"h ⊢ ok (get_tag_name element_ptr) ⟹ element_ptr |∈| element_ptr_kinds h"
assumes get_tag_name_pure [simp]:
"pure (get_tag_name element_ptr) h"
global_interpretation l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
get_tag_name = l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_tag_name and
get_tag_name_locs = l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_tag_name_locs .
interpretation
i_get_tag_name?: l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_tag_name get_tag_name_locs
apply(unfold_locales)
by (auto simp add: get_tag_name_def get_tag_name_locs_def)
declare l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_tag_name_is_l_get_tag_name [instances]:
"l_get_tag_name type_wf get_tag_name get_tag_name_locs"
apply(unfold_locales)
using get_tag_name_reads get_tag_name_ok get_tag_name_ptr_in_heap get_tag_name_pure
by blast+
paragraph ‹set\_disconnected\_nodes›
locale l_set_disconnected_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_disconnected_nodes_get_tag_name:
"∀w ∈ a_set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ a_get_tag_name_locs ptr'. r h h'))"
by(auto simp add: a_set_disconnected_nodes_locs_def a_get_tag_name_locs_def all_args_def)
end
locale l_set_disconnected_nodes_get_tag_name = l_set_disconnected_nodes + l_get_tag_name +
assumes set_disconnected_nodes_get_tag_name:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
interpretation
i_set_disconnected_nodes_get_tag_name?: l_set_disconnected_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
set_disconnected_nodes set_disconnected_nodes_locs
get_tag_name get_tag_name_locs
by unfold_locales
declare l_set_disconnected_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_tag_name_is_l_set_disconnected_nodes_get_tag_name [instances]:
"l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs
get_tag_name get_tag_name_locs"
using set_disconnected_nodes_is_l_set_disconnected_nodes get_tag_name_is_l_get_tag_name
apply(simp add: l_set_disconnected_nodes_get_tag_name_def l_set_disconnected_nodes_get_tag_name_axioms_def)
using set_disconnected_nodes_get_tag_name
by fast
paragraph ‹set\_child\_nodes›
locale l_set_child_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_child_nodes_get_tag_name:
"∀w ∈ set_child_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
by(auto simp add: set_child_nodes_locs_def get_tag_name_locs_def all_args_def
intro: element_put_get_preserved[where getter=tag_name and setter=child_nodes_update])
end
locale l_set_child_nodes_get_tag_name = l_set_child_nodes + l_get_tag_name +
assumes set_child_nodes_get_tag_name:
"∀w ∈ set_child_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
interpretation
i_set_child_nodes_get_tag_name?: l_set_child_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr
set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs
by unfold_locales
declare l_set_child_nodes_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_child_nodes_get_tag_name_is_l_set_child_nodes_get_tag_name [instances]:
"l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs"
using set_child_nodes_is_l_set_child_nodes get_tag_name_is_l_get_tag_name
apply(simp add: l_set_child_nodes_get_tag_name_def l_set_child_nodes_get_tag_name_axioms_def)
using set_child_nodes_get_tag_name
by fast
subsubsection ‹set\_tag\_type›
locale l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_set_tag_name :: "(_) element_ptr ⇒ tag_name ⇒ (_, unit) dom_prog"
where
"a_set_tag_name ptr tag = do {
m ← get_M ptr attrs;
put_M ptr tag_name_update tag
}"
lemmas set_tag_name_defs = a_set_tag_name_def
definition a_set_tag_name_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set"
where
"a_set_tag_name_locs element_ptr ≡ all_args (put_M element_ptr tag_name_update)"
end
locale l_set_tag_name_defs =
fixes set_tag_name :: "(_) element_ptr ⇒ tag_name ⇒ (_, unit) dom_prog"
fixes set_tag_name_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set"
locale l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_set_tag_name_defs set_tag_name set_tag_name_locs +
l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and set_tag_name :: "(_) element_ptr ⇒ char list ⇒ (_, unit) dom_prog"
and set_tag_name_locs :: "(_) element_ptr ⇒ (_, unit) dom_prog set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes set_tag_name_impl: "set_tag_name = a_set_tag_name"
assumes set_tag_name_locs_impl: "set_tag_name_locs = a_set_tag_name_locs"
begin
lemma set_tag_name_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (set_tag_name element_ptr tag)"
apply(unfold type_wf_impl)
unfolding set_tag_name_impl[unfolded a_set_tag_name_def] using get_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok put_M⇩E⇩l⇩e⇩m⇩e⇩n⇩t_ok
by (metis (no_types, lifting) DocumentClass.type_wf⇩E⇩l⇩e⇩m⇩e⇩n⇩t ElementMonad.get_M_pure bind_is_OK_E
bind_is_OK_pure_I is_OK_returns_result_I)
lemma set_tag_name_writes:
"writes (set_tag_name_locs element_ptr) (set_tag_name element_ptr tag) h h'"
by(auto simp add: set_tag_name_impl[unfolded a_set_tag_name_def]
set_tag_name_locs_impl[unfolded a_set_tag_name_locs_def] intro: writes_bind_pure)
lemma set_tag_name_pointers_preserved:
assumes "w ∈ set_tag_name_locs element_ptr"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: all_args_def set_tag_name_locs_impl[unfolded a_set_tag_name_locs_def]
split: if_splits)
lemma set_tag_name_typess_preserved:
assumes "w ∈ set_tag_name_locs element_ptr"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
apply(unfold type_wf_impl)
using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: all_args_def set_tag_name_locs_impl[unfolded a_set_tag_name_locs_def]
split: if_splits)
end
locale l_set_tag_name = l_type_wf + l_set_tag_name_defs +
assumes set_tag_name_writes:
"writes (set_tag_name_locs element_ptr) (set_tag_name element_ptr tag) h h'"
assumes set_tag_name_ok:
"type_wf h ⟹ element_ptr |∈| element_ptr_kinds h ⟹ h ⊢ ok (set_tag_name element_ptr tag)"
assumes set_tag_name_pointers_preserved:
"w ∈ set_tag_name_locs element_ptr ⟹ h ⊢ w →⇩h h' ⟹ object_ptr_kinds h = object_ptr_kinds h'"
assumes set_tag_name_types_preserved:
"w ∈ set_tag_name_locs element_ptr ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
global_interpretation l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
set_tag_name = l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_tag_name and
set_tag_name_locs = l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_tag_name_locs .
interpretation
i_set_tag_name?: l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_tag_name set_tag_name_locs
apply(unfold_locales)
by (auto simp add: set_tag_name_def set_tag_name_locs_def)
declare l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_tag_name_is_l_set_tag_name [instances]:
"l_set_tag_name type_wf set_tag_name set_tag_name_locs"
apply(simp add: l_set_tag_name_def)
using set_tag_name_ok set_tag_name_writes set_tag_name_pointers_preserved
set_tag_name_typess_preserved
by blast
paragraph ‹get\_child\_nodes›
locale l_set_tag_name_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_tag_name_get_child_nodes:
"∀w ∈ set_tag_name_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
by(auto simp add: set_tag_name_locs_impl[unfolded a_set_tag_name_locs_def]
get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def
intro: element_put_get_preserved[where setter=tag_name_update and getter=child_nodes])
end
locale l_set_tag_name_get_child_nodes = l_set_tag_name + l_get_child_nodes +
assumes set_tag_name_get_child_nodes:
"∀w ∈ set_tag_name_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
interpretation
i_set_tag_name_get_child_nodes?: l_set_tag_name_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
set_tag_name set_tag_name_locs known_ptr
get_child_nodes get_child_nodes_locs
by unfold_locales
declare l_set_tag_name_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_tag_name_get_child_nodes_is_l_set_tag_name_get_child_nodes [instances]:
"l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes
get_child_nodes_locs"
using set_tag_name_is_l_set_tag_name get_child_nodes_is_l_get_child_nodes
apply(simp add: l_set_tag_name_get_child_nodes_def l_set_tag_name_get_child_nodes_axioms_def)
using set_tag_name_get_child_nodes
by fast
paragraph ‹get\_disconnected\_nodes›
locale l_set_tag_name_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_tag_name_get_disconnected_nodes:
"∀w ∈ set_tag_name_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
by(auto simp add: set_tag_name_locs_impl[unfolded a_set_tag_name_locs_def]
get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def]
all_args_def)
end
locale l_set_tag_name_get_disconnected_nodes = l_set_tag_name + l_get_disconnected_nodes +
assumes set_tag_name_get_disconnected_nodes:
"∀w ∈ set_tag_name_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
interpretation
i_set_tag_name_get_disconnected_nodes?: l_set_tag_name_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf
set_tag_name set_tag_name_locs get_disconnected_nodes
get_disconnected_nodes_locs
by unfold_locales
declare l_set_tag_name_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_tag_name_get_disconnected_nodes_is_l_set_tag_name_get_disconnected_nodes [instances]:
"l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes
get_disconnected_nodes_locs"
using set_tag_name_is_l_set_tag_name get_disconnected_nodes_is_l_get_disconnected_nodes
apply(simp add: l_set_tag_name_get_disconnected_nodes_def
l_set_tag_name_get_disconnected_nodes_axioms_def)
using set_tag_name_get_disconnected_nodes
by fast
paragraph ‹get\_tag\_type›
locale l_set_tag_name_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M = l_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
+ l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_tag_name_get_tag_name:
assumes "h ⊢ a_set_tag_name element_ptr tag →⇩h h'"
shows "h' ⊢ a_get_tag_name element_ptr →⇩r tag"
using assms
by(auto simp add: a_get_tag_name_def a_set_tag_name_def)
lemma set_tag_name_get_tag_name_different_pointers:
assumes "ptr ≠ ptr'"
assumes "w ∈ a_set_tag_name_locs ptr"
assumes "h ⊢ w →⇩h h'"
assumes "r ∈ a_get_tag_name_locs ptr'"
shows "r h h'"
using assms
by(auto simp add: all_args_def a_set_tag_name_locs_def a_get_tag_name_locs_def
split: if_splits option.splits )
end
locale l_set_tag_name_get_tag_name = l_get_tag_name + l_set_tag_name +
assumes set_tag_name_get_tag_name:
"h ⊢ set_tag_name element_ptr tag →⇩h h'
⟹ h' ⊢ get_tag_name element_ptr →⇩r tag"
assumes set_tag_name_get_tag_name_different_pointers:
"ptr ≠ ptr' ⟹ w ∈ set_tag_name_locs ptr ⟹ h ⊢ w →⇩h h'
⟹ r ∈ get_tag_name_locs ptr' ⟹ r h h'"
interpretation i_set_tag_name_get_tag_name?:
l_set_tag_name_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_tag_name
get_tag_name_locs set_tag_name set_tag_name_locs
by unfold_locales
declare l_set_tag_name_get_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_tag_name_get_tag_name_is_l_set_tag_name_get_tag_name [instances]:
"l_set_tag_name_get_tag_name type_wf get_tag_name get_tag_name_locs
set_tag_name set_tag_name_locs"
using set_tag_name_is_l_set_tag_name get_tag_name_is_l_get_tag_name
apply(simp add: l_set_tag_name_get_tag_name_def
l_set_tag_name_get_tag_name_axioms_def)
using set_tag_name_get_tag_name
set_tag_name_get_tag_name_different_pointers
by fast+
subsubsection ‹set\_val›
locale l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_set_val :: "(_) character_data_ptr ⇒ DOMString ⇒ (_, unit) dom_prog"
where
"a_set_val ptr v = do {
m ← get_M ptr val;
put_M ptr val_update v
}"
lemmas set_val_defs = a_set_val_def
definition a_set_val_locs :: "(_) character_data_ptr ⇒ (_, unit) dom_prog set"
where
"a_set_val_locs character_data_ptr ≡ all_args (put_M character_data_ptr val_update)"
end
locale l_set_val_defs =
fixes set_val :: "(_) character_data_ptr ⇒ DOMString ⇒ (_, unit) dom_prog"
fixes set_val_locs :: "(_) character_data_ptr ⇒ (_, unit) dom_prog set"
locale l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_type_wf type_wf +
l_set_val_defs set_val set_val_locs +
l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
for type_wf :: "(_) heap ⇒ bool"
and set_val :: "(_) character_data_ptr ⇒ char list ⇒ (_, unit) dom_prog"
and set_val_locs :: "(_) character_data_ptr ⇒ (_, unit) dom_prog set" +
assumes type_wf_impl: "type_wf = DocumentClass.type_wf"
assumes set_val_impl: "set_val = a_set_val"
assumes set_val_locs_impl: "set_val_locs = a_set_val_locs"
begin
lemma set_val_ok:
"type_wf h ⟹ character_data_ptr |∈| character_data_ptr_kinds h ⟹ h ⊢ ok (set_val character_data_ptr tag)"
apply(unfold type_wf_impl)
unfolding set_val_impl[unfolded a_set_val_def] using get_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ok put_M⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a_ok
by (metis (no_types, lifting) DocumentClass.type_wf⇩C⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩D⇩a⇩t⇩a CharacterDataMonad.get_M_pure
bind_is_OK_E bind_is_OK_pure_I is_OK_returns_result_I)
lemma set_val_writes: "writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'"
by(auto simp add: set_val_impl[unfolded a_set_val_def] set_val_locs_impl[unfolded a_set_val_locs_def]
intro: writes_bind_pure)
lemma set_val_pointers_preserved:
assumes "w ∈ set_val_locs character_data_ptr"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: all_args_def set_val_locs_impl[unfolded a_set_val_locs_def] split: if_splits)
lemma set_val_typess_preserved:
assumes "w ∈ set_val_locs character_data_ptr"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
apply(unfold type_wf_impl)
using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
by(auto simp add: all_args_def set_val_locs_impl[unfolded a_set_val_locs_def] split: if_splits)
end
locale l_set_val = l_type_wf + l_set_val_defs +
assumes set_val_writes:
"writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'"
assumes set_val_ok:
"type_wf h ⟹ character_data_ptr |∈| character_data_ptr_kinds h ⟹ h ⊢ ok (set_val character_data_ptr tag)"
assumes set_val_pointers_preserved:
"w ∈ set_val_locs character_data_ptr ⟹ h ⊢ w →⇩h h' ⟹ object_ptr_kinds h = object_ptr_kinds h'"
assumes set_val_types_preserved:
"w ∈ set_val_locs character_data_ptr ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
global_interpretation l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs defines
set_val = l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_val and
set_val_locs = l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_set_val_locs .
interpretation
i_set_val?: l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_val set_val_locs
apply(unfold_locales)
by (auto simp add: set_val_def set_val_locs_def)
declare l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_val_is_l_set_val [instances]: "l_set_val type_wf set_val set_val_locs"
apply(simp add: l_set_val_def)
using set_val_ok set_val_writes set_val_pointers_preserved set_val_typess_preserved
by blast
paragraph ‹get\_child\_nodes›
locale l_set_val_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_val_get_child_nodes:
"∀w ∈ set_val_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
by(auto simp add: set_val_locs_impl[unfolded a_set_val_locs_def]
get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def)
end
locale l_set_val_get_child_nodes = l_set_val + l_get_child_nodes +
assumes set_val_get_child_nodes:
"∀w ∈ set_val_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
interpretation
i_set_val_get_child_nodes?: l_set_val_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_val set_val_locs known_ptr
get_child_nodes get_child_nodes_locs
by unfold_locales
declare l_set_val_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_val_get_child_nodes_is_l_set_val_get_child_nodes [instances]:
"l_set_val_get_child_nodes type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs"
using set_val_is_l_set_val get_child_nodes_is_l_get_child_nodes
apply(simp add: l_set_val_get_child_nodes_def l_set_val_get_child_nodes_axioms_def)
using set_val_get_child_nodes
by fast
paragraph ‹get\_disconnected\_nodes›
locale l_set_val_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma set_val_get_disconnected_nodes:
"∀w ∈ set_val_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
by(auto simp add: set_val_locs_impl[unfolded a_set_val_locs_def]
get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def]
all_args_def)
end
locale l_set_val_get_disconnected_nodes = l_set_val + l_get_disconnected_nodes +
assumes set_val_get_disconnected_nodes:
"∀w ∈ set_val_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_disconnected_nodes_locs ptr'. r h h'))"
interpretation
i_set_val_get_disconnected_nodes?: l_set_val_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_val
set_val_locs get_disconnected_nodes get_disconnected_nodes_locs
by unfold_locales
declare l_set_val_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_val_get_disconnected_nodes_is_l_set_val_get_disconnected_nodes [instances]:
"l_set_val_get_disconnected_nodes type_wf set_val set_val_locs get_disconnected_nodes
get_disconnected_nodes_locs"
using set_val_is_l_set_val get_disconnected_nodes_is_l_get_disconnected_nodes
apply(simp add: l_set_val_get_disconnected_nodes_def l_set_val_get_disconnected_nodes_axioms_def)
using set_val_get_disconnected_nodes
by fast
subsubsection ‹get\_parent›
locale l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_child_nodes_defs get_child_nodes get_child_nodes_locs
for get_child_nodes :: "(_::linorder) object_ptr ⇒ (_, (_) node_ptr list) dom_prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
definition a_get_parent :: "(_) node_ptr ⇒ (_, (_::linorder) object_ptr option) dom_prog"
where
"a_get_parent node_ptr = do {
check_in_heap (cast node_ptr);
parent_ptrs ← object_ptr_kinds_M ⤜ filter_M (λptr. do {
children ← get_child_nodes ptr;
return (node_ptr ∈ set children)
});
(if parent_ptrs = []
then return None
else return (Some (hd parent_ptrs)))
}"
definition
"a_get_parent_locs ≡ (⋃ptr. get_child_nodes_locs ptr ∪ {preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr RObject.nothing)})"
end
locale l_get_parent_defs =
fixes get_parent :: "(_) node_ptr ⇒ (_, (_::linorder) object_ptr option) dom_prog"
fixes get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs +
l_known_ptrs known_ptr known_ptrs +
l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs +
l_get_parent_defs get_parent get_parent_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes
and get_child_nodes_locs
and known_ptrs :: "(_) heap ⇒ bool"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs +
assumes get_parent_impl: "get_parent = a_get_parent"
assumes get_parent_locs_impl: "get_parent_locs = a_get_parent_locs"
begin
lemmas get_parent_def = get_parent_impl[unfolded a_get_parent_def]
lemmas get_parent_locs_def = get_parent_locs_impl[unfolded a_get_parent_locs_def]
lemma get_parent_pure [simp]: "pure (get_parent ptr) h"
using get_child_nodes_pure
by(auto simp add: get_parent_def intro!: bind_pure_I filter_M_pure_I)
lemma get_parent_ok [simp]:
assumes "type_wf h"
assumes "known_ptrs h"
assumes "ptr |∈| node_ptr_kinds h"
shows "h ⊢ ok (get_parent ptr)"
using assms get_child_nodes_ok get_child_nodes_pure
by(auto simp add: get_parent_impl[unfolded a_get_parent_def] known_ptrs_known_ptr
intro!: bind_is_OK_pure_I filter_M_pure_I filter_M_is_OK_I bind_pure_I)
lemma get_parent_ptr_in_heap [simp]: "h ⊢ ok (get_parent node_ptr) ⟹ node_ptr |∈| node_ptr_kinds h"
using get_parent_def is_OK_returns_result_I check_in_heap_ptr_in_heap
by (metis (no_types, lifting) bind_returns_heap_E get_parent_pure node_ptr_kinds_commutes pure_pure)
lemma get_parent_parent_in_heap:
assumes "h ⊢ get_parent child_node →⇩r Some parent"
shows "parent |∈| object_ptr_kinds h"
using assms get_child_nodes_pure
by(auto simp add: get_parent_def elim!: bind_returns_result_E2
dest!: filter_M_not_more_elements[where x=parent]
intro!: filter_M_pure_I bind_pure_I
split: if_splits)
lemma get_parent_child_dual:
assumes "h ⊢ get_parent child →⇩r Some ptr"
obtains children where "h ⊢ get_child_nodes ptr →⇩r children" and "child ∈ set children"
using assms get_child_nodes_pure
by(auto simp add: get_parent_def bind_pure_I
dest!: filter_M_holds_for_result
elim!: bind_returns_result_E2
intro!: filter_M_pure_I
split: if_splits)
lemma get_parent_reads: "reads get_parent_locs (get_parent node_ptr) h h'"
using get_child_nodes_reads[unfolded reads_def]
by(auto simp add: get_parent_def get_parent_locs_def
intro!: reads_bind_pure reads_subset[OF check_in_heap_reads]
reads_subset[OF get_child_nodes_reads] reads_subset[OF return_reads]
reads_subset[OF object_ptr_kinds_M_reads] filter_M_reads filter_M_pure_I bind_pure_I)
lemma get_parent_reads_pointers: "preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr RObject.nothing) ∈ get_parent_locs"
by(auto simp add: get_parent_locs_def)
end
locale l_get_parent = l_type_wf + l_known_ptrs + l_get_parent_defs + l_get_child_nodes +
assumes get_parent_reads:
"reads get_parent_locs (get_parent node_ptr) h h'"
assumes get_parent_ok:
"type_wf h ⟹ known_ptrs h ⟹ node_ptr |∈| node_ptr_kinds h ⟹ h ⊢ ok (get_parent node_ptr)"
assumes get_parent_ptr_in_heap:
"h ⊢ ok (get_parent node_ptr) ⟹ node_ptr |∈| node_ptr_kinds h"
assumes get_parent_pure [simp]:
"pure (get_parent node_ptr) h"
assumes get_parent_parent_in_heap:
"h ⊢ get_parent child_node →⇩r Some parent ⟹ parent |∈| object_ptr_kinds h"
assumes get_parent_child_dual:
"h ⊢ get_parent child →⇩r Some ptr ⟹ (⋀children. h ⊢ get_child_nodes ptr →⇩r children
⟹ child ∈ set children ⟹ thesis) ⟹ thesis"
assumes get_parent_reads_pointers:
"preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr RObject.nothing) ∈ get_parent_locs"
global_interpretation l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs defines
get_parent = "l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_parent get_child_nodes" and
get_parent_locs = "l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_parent_locs get_child_nodes_locs" .
interpretation
i_get_parent?: l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs
get_parent get_parent_locs
using instances
apply(simp add: l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def)
apply(simp add: get_parent_def get_parent_locs_def)
done
declare l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_parent_is_l_get_parent [instances]:
"l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs"
using instances
apply(auto simp add: l_get_parent_def l_get_parent_axioms_def)[1]
using get_parent_reads get_parent_ok get_parent_ptr_in_heap get_parent_pure
get_parent_parent_in_heap get_parent_child_dual
using get_parent_reads_pointers
by blast+
paragraph ‹set\_disconnected\_nodes›
locale l_set_disconnected_nodes_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_disconnected_nodes_get_child_nodes
set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs
+ l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf set_disconnected_nodes set_disconnected_nodes_locs
+ l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and known_ptrs :: "(_) heap ⇒ bool"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma set_disconnected_nodes_get_parent [simp]:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_parent_locs. r h h'))"
by(auto simp add: get_parent_locs_def set_disconnected_nodes_locs_def all_args_def)
end
locale l_set_disconnected_nodes_get_parent = l_set_disconnected_nodes_defs + l_get_parent_defs +
assumes set_disconnected_nodes_get_parent [simp]:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_parent_locs. r h h'))"
interpretation i_set_disconnected_nodes_get_parent?:
l_set_disconnected_nodes_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf set_disconnected_nodes
set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
using instances
by (simp add: l_set_disconnected_nodes_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_set_disconnected_nodes_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_parent_is_l_set_disconnected_nodes_get_parent [instances]:
"l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"
by(simp add: l_set_disconnected_nodes_get_parent_def)
subsubsection ‹get\_root\_node›
locale l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_parent_defs get_parent get_parent_locs
for get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_::linorder) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
begin
partial_function (dom_prog)
a_get_ancestors :: "(_::linorder) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
where
"a_get_ancestors ptr = do {
check_in_heap ptr;
ancestors ← (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ptr of
Some node_ptr ⇒ do {
parent_ptr_opt ← get_parent node_ptr;
(case parent_ptr_opt of
Some parent_ptr ⇒ a_get_ancestors parent_ptr
| None ⇒ return [])
}
| None ⇒ return []);
return (ptr # ancestors)
}"
definition "a_get_ancestors_locs = get_parent_locs"
definition a_get_root_node :: "(_) object_ptr ⇒ (_, (_) object_ptr) dom_prog"
where
"a_get_root_node ptr = do {
ancestors ← a_get_ancestors ptr;
return (last ancestors)
}"
definition "a_get_root_node_locs = a_get_ancestors_locs"
end
locale l_get_ancestors_defs =
fixes get_ancestors :: "(_::linorder) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
fixes get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_root_node_defs =
fixes get_root_node :: "(_) object_ptr ⇒ (_, (_) object_ptr) dom_prog"
fixes get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
locale l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_parent +
l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs +
l_get_ancestors_defs +
l_get_root_node_defs +
assumes get_ancestors_impl: "get_ancestors = a_get_ancestors"
assumes get_ancestors_locs_impl: "get_ancestors_locs = a_get_ancestors_locs"
assumes get_root_node_impl: "get_root_node = a_get_root_node"
assumes get_root_node_locs_impl: "get_root_node_locs = a_get_root_node_locs"
begin
lemmas get_ancestors_def = a_get_ancestors.simps[folded get_ancestors_impl]
lemmas get_ancestors_locs_def = a_get_ancestors_locs_def[folded get_ancestors_locs_impl]
lemmas get_root_node_def = a_get_root_node_def[folded get_root_node_impl get_ancestors_impl]
lemmas get_root_node_locs_def = a_get_root_node_locs_def[folded get_root_node_locs_impl
get_ancestors_locs_impl]
lemma get_ancestors_pure [simp]:
"pure (get_ancestors ptr) h"
proof -
have "∀ptr h h' x. h ⊢ get_ancestors ptr →⇩r x ⟶ h ⊢ get_ancestors ptr →⇩h h' ⟶ h = h'"
proof (induct rule: a_get_ancestors.fixp_induct[folded get_ancestors_impl])
case 1
then show ?case
by(rule admissible_dom_prog)
next
case 2
then show ?case
by simp
next
case (3 f)
then show ?case
using get_parent_pure
apply(auto simp add: pure_returns_heap_eq pure_def
split: option.splits
elim!: bind_returns_heap_E bind_returns_result_E
dest!: pure_returns_heap_eq[rotated, OF check_in_heap_pure])[1]
apply (meson option.simps(3) returns_result_eq)
by (metis get_parent_pure pure_returns_heap_eq)
qed
then show ?thesis
by (meson pure_eq_iff)
qed
lemma get_root_node_pure [simp]: "pure (get_root_node ptr) h"
by(auto simp add: get_root_node_def bind_pure_I)
lemma get_ancestors_ptr_in_heap:
assumes "h ⊢ ok (get_ancestors ptr)"
shows "ptr |∈| object_ptr_kinds h"
using assms
by(auto simp add: get_ancestors_def check_in_heap_ptr_in_heap
elim!: bind_is_OK_E dest: is_OK_returns_result_I)
lemma get_ancestors_ptr:
assumes "h ⊢ get_ancestors ptr →⇩r ancestors"
shows "ptr ∈ set ancestors"
using assms
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits intro!: bind_pure_I)
lemma get_ancestors_not_node:
assumes "h ⊢ get_ancestors ptr →⇩r ancestors"
assumes "¬is_node_ptr_kind ptr"
shows "ancestors = [ptr]"
using assms
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits)
lemma get_root_node_no_parent:
"h ⊢ get_parent node_ptr →⇩r None ⟹ h ⊢ get_root_node (cast node_ptr) →⇩r cast node_ptr"
apply(auto simp add: check_in_heap_def get_root_node_def get_ancestors_def
intro!: bind_pure_returns_result_I )[1]
using get_parent_ptr_in_heap by blast
end
locale l_get_ancestors = l_get_ancestors_defs +
assumes get_ancestors_pure [simp]: "pure (get_ancestors node_ptr) h"
assumes get_ancestors_ptr_in_heap: "h ⊢ ok (get_ancestors ptr) ⟹ ptr |∈| object_ptr_kinds h"
assumes get_ancestors_ptr: "h ⊢ get_ancestors ptr →⇩r ancestors ⟹ ptr ∈ set ancestors"
locale l_get_root_node = l_get_root_node_defs + l_get_parent_defs +
assumes get_root_node_pure[simp]:
"pure (get_root_node ptr) h"
assumes get_root_node_no_parent:
"h ⊢ get_parent node_ptr →⇩r None ⟹ h ⊢ get_root_node (cast node_ptr) →⇩r cast node_ptr"
global_interpretation l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_parent get_parent_locs
defines get_root_node = "l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_root_node get_parent"
and get_root_node_locs = "l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_root_node_locs get_parent_locs"
and get_ancestors = "l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_ancestors get_parent"
and get_ancestors_locs = "l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_ancestors_locs get_parent_locs"
.
declare a_get_ancestors.simps [code]
interpretation
i_get_root_node?: l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf known_ptr known_ptrs get_parent get_parent_locs
get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs
get_root_node get_root_node_locs
using instances
apply(simp add: l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def)
by(simp add: get_root_node_def get_root_node_locs_def get_ancestors_def get_ancestors_locs_def)
declare l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_ancestors_is_l_get_ancestors [instances]: "l_get_ancestors get_ancestors"
unfolding l_get_ancestors_def
using get_ancestors_pure get_ancestors_ptr get_ancestors_ptr_in_heap
by blast
lemma get_root_node_is_l_get_root_node [instances]: "l_get_root_node get_root_node get_parent"
apply(simp add: l_get_root_node_def)
using get_root_node_no_parent
by fast
paragraph ‹set\_disconnected\_nodes›
locale l_set_disconnected_nodes_get_ancestors⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_disconnected_nodes_get_parent
set_disconnected_nodes set_disconnected_nodes_locs get_parent get_parent_locs
+ l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs
get_ancestors get_ancestors_locs get_root_node get_root_node_locs
+ l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf set_disconnected_nodes set_disconnected_nodes_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and type_wf :: "(_) heap ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma set_disconnected_nodes_get_ancestors:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_ancestors_locs. r h h'))"
by(auto simp add: get_parent_locs_def set_disconnected_nodes_locs_def get_ancestors_locs_def
all_args_def)
end
locale l_set_disconnected_nodes_get_ancestors = l_set_disconnected_nodes_defs + l_get_ancestors_defs +
assumes set_disconnected_nodes_get_ancestors:
"∀w ∈ set_disconnected_nodes_locs ptr. (h ⊢ w →⇩h h' ⟶ (∀r ∈ get_ancestors_locs. r h h'))"
interpretation
i_set_disconnected_nodes_get_ancestors?: l_set_disconnected_nodes_get_ancestors⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr
set_disconnected_nodes set_disconnected_nodes_locs
get_child_nodes get_child_nodes_locs get_parent
get_parent_locs type_wf known_ptrs get_ancestors
get_ancestors_locs get_root_node get_root_node_locs
using instances
by (simp add: l_set_disconnected_nodes_get_ancestors⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_set_disconnected_nodes_get_ancestors⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_ancestors_is_l_set_disconnected_nodes_get_ancestors [instances]:
"l_set_disconnected_nodes_get_ancestors set_disconnected_nodes_locs get_ancestors_locs"
using instances
apply(simp add: l_set_disconnected_nodes_get_ancestors_def)
using set_disconnected_nodes_get_ancestors
by fast
subsubsection ‹get\_owner\_document›
locale l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_get_root_node_defs get_root_node get_root_node_locs
for get_root_node :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
definition a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r :: "(_) node_ptr ⇒ unit ⇒ (_, (_) document_ptr) dom_prog"
where
"a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr _ = do {
root ← get_root_node (cast node_ptr);
(case cast root of
Some document_ptr ⇒ return document_ptr
| None ⇒ do {
ptrs ← document_ptr_kinds_M;
candidates ← filter_M (λdocument_ptr. do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (root ∈ cast ` set disconnected_nodes)
}) ptrs;
return (hd candidates)
})
}"
definition
a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r :: "(_) document_ptr ⇒ unit ⇒ (_, (_) document_ptr) dom_prog"
where
"a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr _ = do {
document_ptrs ← document_ptr_kinds_M;
(if document_ptr ∈ set document_ptrs then return document_ptr else error SegmentationFault)}"
definition
a_get_owner_document_tups :: "(((_) object_ptr ⇒ bool) × ((_) object_ptr ⇒ unit
⇒ (_, (_) document_ptr) dom_prog)) list"
where
"a_get_owner_document_tups = [
(is_element_ptr, a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_character_data_ptr, a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ∘ the ∘ cast),
(is_document_ptr, a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast)
]"
definition a_get_owner_document :: "(_) object_ptr ⇒ (_, (_) document_ptr) dom_prog"
where
"a_get_owner_document ptr = invoke a_get_owner_document_tups ptr ()"
end
locale l_get_owner_document_defs =
fixes get_owner_document :: "(_::linorder) object_ptr ⇒ (_, (_) document_ptr) dom_prog"
locale l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_known_ptr known_ptr +
l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs +
l_get_root_node get_root_node get_root_node_locs +
l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_root_node get_root_node_locs get_disconnected_nodes
get_disconnected_nodes_locs +
l_get_owner_document_defs get_owner_document
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_owner_document :: "(_) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog" +
assumes known_ptr_impl: "known_ptr = a_known_ptr"
assumes get_owner_document_impl: "get_owner_document = a_get_owner_document"
begin
lemmas known_ptr_def = known_ptr_impl[unfolded a_known_ptr_def]
lemmas get_owner_document_def = a_get_owner_document_def[folded get_owner_document_impl]
lemma get_owner_document_split:
"P (invoke (a_get_owner_document_tups @ xs) ptr ()) =
((known_ptr ptr ⟶ P (get_owner_document ptr))
∧ (¬(known_ptr ptr) ⟶ P (invoke xs ptr ())))"
by(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_def
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs
NodeClass.known_ptr_defs
split: invoke_splits option.splits)
lemma get_owner_document_split_asm:
"P (invoke (a_get_owner_document_tups @ xs) ptr ()) =
(¬((known_ptr ptr ∧ ¬P (get_owner_document ptr))
∨ (¬(known_ptr ptr) ∧ ¬P (invoke xs ptr ()))))"
by(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_def
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs
NodeClass.known_ptr_defs
split: invoke_splits)
lemmas get_owner_document_splits = get_owner_document_split get_owner_document_split_asm
lemma get_owner_document_pure [simp]:
"pure (get_owner_document ptr) h"
proof -
have "⋀node_ptr. pure (a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr ()) h"
by(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_pure_I filter_M_pure_I
split: option.splits)
moreover have "⋀document_ptr. pure (a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr ()) h"
by(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def bind_pure_I)
ultimately show ?thesis
by(auto simp add: get_owner_document_def a_get_owner_document_tups_def
intro!: bind_pure_I
split: invoke_splits)
qed
lemma get_owner_document_ptr_in_heap:
assumes "h ⊢ ok (get_owner_document ptr)"
shows "ptr |∈| object_ptr_kinds h"
using assms
by(auto simp add: get_owner_document_def invoke_ptr_in_heap dest: is_OK_returns_heap_I)
end
locale l_get_owner_document = l_get_owner_document_defs +
assumes get_owner_document_ptr_in_heap:
"h ⊢ ok (get_owner_document ptr) ⟹ ptr |∈| object_ptr_kinds h"
assumes get_owner_document_pure [simp]:
"pure (get_owner_document ptr) h"
global_interpretation l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_root_node get_root_node_locs
get_disconnected_nodes get_disconnected_nodes_locs
defines get_owner_document_tups =
"l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_owner_document_tups get_root_node get_disconnected_nodes"
and get_owner_document =
"l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_owner_document get_root_node get_disconnected_nodes"
and get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r =
"l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r get_root_node get_disconnected_nodes"
.
interpretation
i_get_owner_document?: l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_parent get_parent_locs known_ptr type_wf
get_disconnected_nodes get_disconnected_nodes_locs get_root_node
get_root_node_locs get_owner_document
using instances
apply(auto simp add: l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def)[1]
by(auto simp add: get_owner_document_tups_def get_owner_document_def get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def)[1]
declare l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_owner_document_is_l_get_owner_document [instances]:
"l_get_owner_document get_owner_document"
using get_owner_document_ptr_in_heap
by(auto simp add: l_get_owner_document_def)
subsubsection ‹remove\_child›
locale l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
l_set_child_nodes_defs set_child_nodes set_child_nodes_locs +
l_get_parent_defs get_parent get_parent_locs +
l_get_owner_document_defs get_owner_document +
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs
for get_child_nodes :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_owner_document :: "(_) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
begin
definition a_remove_child :: "(_) object_ptr ⇒ (_) node_ptr ⇒ (_, unit) dom_prog"
where
"a_remove_child ptr child = do {
children ← get_child_nodes ptr;
if child ∉ set children then
error NotFoundError
else do {
owner_document ← get_owner_document (cast child);
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (child # disc_nodes);
set_child_nodes ptr (remove1 child children)
}
}"
definition a_remove_child_locs :: "(_) object_ptr ⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
where
"a_remove_child_locs ptr owner_document = set_child_nodes_locs ptr
∪ set_disconnected_nodes_locs owner_document"
definition a_remove :: "(_) node_ptr ⇒ (_, unit) dom_prog"
where
"a_remove node_ptr = do {
parent_opt ← get_parent node_ptr;
(case parent_opt of
Some parent ⇒ do {
a_remove_child parent node_ptr;
return ()
}
| None ⇒ return ())
}"
end
locale l_remove_child_defs =
fixes remove_child :: "(_::linorder) object_ptr ⇒ (_) node_ptr ⇒ (_, unit) dom_prog"
fixes remove_child_locs :: "(_) object_ptr ⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
locale l_remove_defs =
fixes remove :: "(_) node_ptr ⇒ (_, unit) dom_prog"
locale l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs +
l_remove_child_defs +
l_remove_defs +
l_get_parent +
l_get_owner_document +
l_set_child_nodes_get_child_nodes +
l_set_child_nodes_get_disconnected_nodes +
l_set_disconnected_nodes_get_disconnected_nodes +
l_set_disconnected_nodes_get_child_nodes +
assumes remove_child_impl: "remove_child = a_remove_child"
assumes remove_child_locs_impl: "remove_child_locs = a_remove_child_locs"
assumes remove_impl: "remove = a_remove"
begin
lemmas remove_child_def = a_remove_child_def[folded remove_child_impl]
lemmas remove_child_locs_def = a_remove_child_locs_def[folded remove_child_locs_impl]
lemmas remove_def = a_remove_def[folded remove_child_impl remove_impl]
lemma remove_child_ptr_in_heap:
assumes "h ⊢ ok (remove_child ptr child)"
shows "ptr |∈| object_ptr_kinds h"
proof -
obtain children where children: "h ⊢ get_child_nodes ptr →⇩r children"
using assms
by(auto simp add: remove_child_def)
moreover have "children ≠ []"
using assms calculation
by(auto simp add: remove_child_def elim!: bind_is_OK_E2)
ultimately show ?thesis
using assms(1) get_child_nodes_ptr_in_heap by blast
qed
lemma remove_child_child_in_heap:
assumes "h ⊢ remove_child ptr' child →⇩h h'"
shows "child |∈| node_ptr_kinds h"
using assms
apply(auto simp add: remove_child_def
elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
split: if_splits)[1]
by (meson is_OK_returns_result_I local.get_owner_document_ptr_in_heap node_ptr_kinds_commutes)
lemma remove_child_in_disconnected_nodes:
assumes "h ⊢ remove_child ptr child →⇩h h'"
assumes "h ⊢ get_owner_document (cast child) →⇩r owner_document"
assumes "h' ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes"
shows "child ∈ set disc_nodes"
proof -
obtain prev_disc_nodes h2 children where
disc_nodes: "h ⊢ get_disconnected_nodes owner_document →⇩r prev_disc_nodes" and
h2: "h ⊢ set_disconnected_nodes owner_document (child # prev_disc_nodes) →⇩h h2" and
h': "h2 ⊢ set_child_nodes ptr (remove1 child children) →⇩h h'"
using assms(1)
apply(auto simp add: remove_child_def
elim!: bind_returns_heap_E
dest!: returns_result_eq[OF assms(2)]
pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_child_nodes_pure]
split: if_splits)[1]
by (metis get_disconnected_nodes_pure pure_returns_heap_eq)
have "h2 ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes"
apply(rule reads_writes_separate_backwards[OF get_disconnected_nodes_reads
set_child_nodes_writes h' assms(3)])
by (simp add: set_child_nodes_get_disconnected_nodes)
then show ?thesis
by (metis (no_types, lifting) h2 set_disconnected_nodes_get_disconnected_nodes
list.set_intros(1) select_result_I2)
qed
lemma remove_child_writes [simp]:
"writes (remove_child_locs ptr |h ⊢ get_owner_document (cast child)|⇩r) (remove_child ptr child) h h'"
apply(auto simp add: remove_child_def intro!: writes_bind_pure[OF get_child_nodes_pure]
writes_bind_pure[OF get_owner_document_pure]
writes_bind_pure[OF get_disconnected_nodes_pure])[1]
by(auto simp add: remove_child_locs_def set_disconnected_nodes_writes writes_union_right_I
set_child_nodes_writes writes_union_left_I
intro!: writes_bind)
lemma remove_writes:
"writes (remove_child_locs (the |h ⊢ get_parent child|⇩r) |h ⊢ get_owner_document (cast child)|⇩r)
(remove child) h h'"
by(auto simp add: remove_def intro!: writes_bind_pure split: option.splits)
lemma remove_child_children_subset:
assumes "h ⊢ remove_child parent child →⇩h h'"
and "h ⊢ get_child_nodes ptr →⇩r children"
and "h' ⊢ get_child_nodes ptr →⇩r children'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "set children' ⊆ set children"
proof -
obtain ptr_children owner_document h2 disc_nodes where
owner_document: "h ⊢ get_owner_document (cast child) →⇩r owner_document" and
ptr_children: "h ⊢ get_child_nodes parent →⇩r ptr_children" and
disc_nodes: "h ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h2: "h ⊢ set_disconnected_nodes owner_document (child # disc_nodes) →⇩h h2" and
h': "h2 ⊢ set_child_nodes parent (remove1 child ptr_children) →⇩h h'"
using assms(1)
by(auto simp add: remove_child_def
elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_disconnected_nodes_pure]
pure_returns_heap_eq[rotated, OF get_child_nodes_pure]
split: if_splits)
have "parent |∈| object_ptr_kinds h"
using get_child_nodes_ptr_in_heap ptr_children by blast
have "object_ptr_kinds h = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h2])
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
have "type_wf h2"
using type_wf writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF set_disconnected_nodes_writes h2]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have "h2 ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h2 assms(2)
apply(rule reads_writes_separate_forwards)
by (simp add: set_disconnected_nodes_get_child_nodes)
moreover have "h2 ⊢ get_child_nodes parent →⇩r ptr_children"
using get_child_nodes_reads set_disconnected_nodes_writes h2 ptr_children
apply(rule reads_writes_separate_forwards)
by (simp add: set_disconnected_nodes_get_child_nodes)
moreover have
"ptr ≠ parent ⟹ h2 ⊢ get_child_nodes ptr →⇩r children = h' ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_child_nodes_writes h'
apply(rule reads_writes_preserved)
by (metis set_child_nodes_get_child_nodes_different_pointers)
moreover have "h' ⊢ get_child_nodes parent →⇩r remove1 child ptr_children"
using h' set_child_nodes_get_child_nodes known_ptrs type_wf known_ptrs_known_ptr
‹parent |∈| object_ptr_kinds h› ‹object_ptr_kinds h = object_ptr_kinds h2› ‹type_wf h2›
by fast
moreover have "set ( remove1 child ptr_children) ⊆ set ptr_children"
by (simp add: set_remove1_subset)
ultimately show ?thesis
by (metis assms(3) order_refl returns_result_eq)
qed
lemma remove_child_pointers_preserved:
assumes "w ∈ remove_child_locs ptr owner_document"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms
using set_child_nodes_pointers_preserved
using set_disconnected_nodes_pointers_preserved
unfolding remove_child_locs_def
by auto
lemma remove_child_types_preserved:
assumes "w ∈ remove_child_locs ptr owner_document"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
using assms
using set_child_nodes_types_preserved
using set_disconnected_nodes_types_preserved
unfolding remove_child_locs_def
by auto
end
locale l_remove_child = l_type_wf + l_known_ptrs + l_remove_child_defs + l_get_owner_document_defs
+ l_get_child_nodes_defs + l_get_disconnected_nodes_defs +
assumes remove_child_writes:
"writes (remove_child_locs object_ptr |h ⊢ get_owner_document (cast child)|⇩r)
(remove_child object_ptr child) h h'"
assumes remove_child_pointers_preserved:
"w ∈ remove_child_locs ptr owner_document ⟹ h ⊢ w →⇩h h' ⟹ object_ptr_kinds h = object_ptr_kinds h'"
assumes remove_child_types_preserved:
"w ∈ remove_child_locs ptr owner_document ⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
assumes remove_child_in_disconnected_nodes:
"known_ptrs h ⟹ h ⊢ remove_child ptr child →⇩h h'
⟹ h ⊢ get_owner_document (cast child) →⇩r owner_document
⟹ h' ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes
⟹ child ∈ set disc_nodes"
assumes remove_child_ptr_in_heap: "h ⊢ ok (remove_child ptr child) ⟹ ptr |∈| object_ptr_kinds h"
assumes remove_child_child_in_heap: "h ⊢ remove_child ptr' child →⇩h h' ⟹ child |∈| node_ptr_kinds h"
assumes remove_child_children_subset:
"known_ptrs h ⟹ type_wf h ⟹ h ⊢ remove_child parent child →⇩h h'
⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h' ⊢ get_child_nodes ptr →⇩r children'
⟹ set children' ⊆ set children"
locale l_remove
global_interpretation l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs set_child_nodes
set_child_nodes_locs get_parent get_parent_locs
get_owner_document get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs
defines remove =
"l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_remove get_child_nodes set_child_nodes get_parent get_owner_document
get_disconnected_nodes set_disconnected_nodes"
and remove_child =
"l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document
get_disconnected_nodes set_disconnected_nodes"
and remove_child_locs =
"l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs"
.
interpretation
i_remove_child?: l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_child_nodes get_child_nodes_locs set_child_nodes
set_child_nodes_locs get_parent get_parent_locs get_owner_document
get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf
known_ptr known_ptrs
using instances
apply(simp add: l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def)
by(simp add: remove_child_def remove_child_locs_def remove_def)
declare l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma remove_child_is_l_remove_child [instances]:
"l_remove_child type_wf known_ptr known_ptrs remove_child remove_child_locs get_owner_document
get_child_nodes get_disconnected_nodes"
using instances
apply(auto simp add: l_remove_child_def l_remove_child_axioms_def)[1]
using remove_child_pointers_preserved apply(blast)
using remove_child_pointers_preserved apply(blast)
using remove_child_types_preserved apply(blast)
using remove_child_types_preserved apply(blast)
using remove_child_in_disconnected_nodes apply(blast)
using remove_child_ptr_in_heap apply(blast)
using remove_child_child_in_heap apply(blast)
using remove_child_children_subset apply(blast)
done
subsubsection ‹adopt\_node›
locale l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_owner_document_defs get_owner_document +
l_get_parent_defs get_parent get_parent_locs +
l_remove_child_defs remove_child remove_child_locs +
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs
for get_owner_document :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and remove_child :: "(_) object_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and remove_child_locs :: "(_) object_ptr ⇒ (_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
begin
definition a_adopt_node :: "(_) document_ptr ⇒ (_) node_ptr ⇒ (_, unit) dom_prog"
where
"a_adopt_node document node = do {
old_document ← get_owner_document (cast node);
parent_opt ← get_parent node;
(case parent_opt of
Some parent ⇒ do {
remove_child parent node
} | None ⇒ do {
return ()
});
(if document ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 node old_disc_nodes);
disc_nodes ← get_disconnected_nodes document;
set_disconnected_nodes document (node # disc_nodes)
} else do {
return ()
})
}"
definition
a_adopt_node_locs :: "(_) object_ptr option ⇒ (_) document_ptr ⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
where
"a_adopt_node_locs parent owner_document document_ptr =
((if parent = None
then {}
else remove_child_locs (the parent) owner_document) ∪ set_disconnected_nodes_locs document_ptr
∪ set_disconnected_nodes_locs owner_document)"
end
locale l_adopt_node_defs =
fixes
adopt_node :: "(_) document_ptr ⇒ (_) node_ptr ⇒ (_, unit) dom_prog"
fixes
adopt_node_locs :: "(_) object_ptr option ⇒ (_) document_ptr ⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
global_interpretation l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_owner_document get_parent get_parent_locs remove_child
remove_child_locs get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs
defines adopt_node = "l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_adopt_node get_owner_document get_parent remove_child
get_disconnected_nodes set_disconnected_nodes"
and adopt_node_locs = "l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_adopt_node_locs
remove_child_locs set_disconnected_nodes_locs"
.
locale l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs
+ l_adopt_node_defs
adopt_node adopt_node_locs
+ l_get_owner_document
get_owner_document
+ l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
+ l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent
get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf
known_ptr known_ptrs
+ l_set_disconnected_nodes_get_disconnected_nodes
type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs
for get_owner_document :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and remove_child :: "(_) object_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and remove_child_locs :: "(_) object_ptr ⇒ (_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and adopt_node :: "(_) document_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and adopt_node_locs :: "(_) object_ptr option ⇒ (_) document_ptr ⇒ (_) document_ptr
⇒ ((_) heap, exception, unit) prog set"
and known_ptr :: "(_) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and known_ptrs :: "(_) heap ⇒ bool"
and set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap, exception, unit) prog set"
and remove :: "(_) node_ptr ⇒ ((_) heap, exception, unit) prog" +
assumes adopt_node_impl: "adopt_node = a_adopt_node"
assumes adopt_node_locs_impl: "adopt_node_locs = a_adopt_node_locs"
begin
lemmas adopt_node_def = a_adopt_node_def[folded adopt_node_impl]
lemmas adopt_node_locs_def = a_adopt_node_locs_def[folded adopt_node_locs_impl]
lemma adopt_node_writes:
shows "writes (adopt_node_locs |h ⊢ get_parent node|⇩r |h
⊢ get_owner_document (cast node)|⇩r document_ptr) (adopt_node document_ptr node) h h'"
apply(auto simp add: adopt_node_def adopt_node_locs_def
intro!: writes_bind_pure[OF get_owner_document_pure] writes_bind_pure[OF get_parent_pure]
writes_bind_pure[OF get_disconnected_nodes_pure]
split: option.splits)[1]
apply(auto intro!: writes_bind)[1]
apply (simp add: set_disconnected_nodes_writes writes_union_right_I)
apply (simp add: set_disconnected_nodes_writes writes_union_left_I writes_union_right_I)
apply(auto intro!: writes_bind)[1]
apply (metis (no_types, lifting) remove_child_writes select_result_I2 writes_union_left_I)
apply (simp add: set_disconnected_nodes_writes writes_union_right_I)
by(auto intro: writes_subset[OF set_disconnected_nodes_writes] writes_subset[OF remove_child_writes])
lemma adopt_node_children_subset:
assumes "h ⊢ adopt_node owner_document node →⇩h h'"
and "h ⊢ get_child_nodes ptr →⇩r children"
and "h' ⊢ get_child_nodes ptr →⇩r children'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "set children' ⊆ set children"
proof -
obtain old_document parent_opt h2 where
old_document: "h ⊢ get_owner_document (cast node) →⇩r old_document" and
parent_opt: "h ⊢ get_parent node →⇩r parent_opt" and
h2: "h ⊢ (case parent_opt of Some parent ⇒ do { remove_child parent node } |
None ⇒ do { return ()}) →⇩h h2"
and
h': "h2 ⊢ (if owner_document ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 node old_disc_nodes);
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (node # disc_nodes)
} else do { return () }) →⇩h h'"
using assms(1)
by(auto simp add: adopt_node_def
elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_parent_pure])
have "h2 ⊢ get_child_nodes ptr →⇩r children'"
proof (cases "owner_document ≠ old_document")
case True
then obtain h3 old_disc_nodes disc_nodes where
old_disc_nodes: "h2 ⊢ get_disconnected_nodes old_document →⇩r old_disc_nodes" and
h3: "h2 ⊢ set_disconnected_nodes old_document (remove1 node old_disc_nodes) →⇩h h3" and
old_disc_nodes: "h3 ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h': "h3 ⊢ set_disconnected_nodes owner_document (node # disc_nodes) →⇩h h'"
using h'
by(auto elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
have "h3 ⊢ get_child_nodes ptr →⇩r children'"
using get_child_nodes_reads set_disconnected_nodes_writes h' assms(3)
apply(rule reads_writes_separate_backwards)
by (simp add: set_disconnected_nodes_get_child_nodes)
show ?thesis
using get_child_nodes_reads set_disconnected_nodes_writes h3 ‹h3 ⊢ get_child_nodes ptr →⇩r children'›
apply(rule reads_writes_separate_backwards)
by (simp add: set_disconnected_nodes_get_child_nodes)
next
case False
then show ?thesis
using h' assms(3) by(auto)
qed
show ?thesis
proof (insert h2, induct parent_opt)
case None
then show ?case
using assms
by(auto dest!: returns_result_eq[OF ‹h2 ⊢ get_child_nodes ptr →⇩r children'›])
next
case (Some option)
then show ?case
using assms(2) ‹h2 ⊢ get_child_nodes ptr →⇩r children'› remove_child_children_subset known_ptrs
type_wf
by simp
qed
qed
lemma adopt_node_child_in_heap:
assumes "h ⊢ ok (adopt_node document_ptr child)"
shows "child |∈| node_ptr_kinds h"
using assms
apply(auto simp add: adopt_node_def elim!: bind_is_OK_E)[1]
using get_owner_document_pure get_parent_ptr_in_heap pure_returns_heap_eq
by fast
lemma adopt_node_pointers_preserved:
assumes "w ∈ adopt_node_locs parent owner_document document_ptr"
assumes "h ⊢ w →⇩h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
using assms
using set_disconnected_nodes_pointers_preserved
using remove_child_pointers_preserved
unfolding adopt_node_locs_def
by (auto split: if_splits)
lemma adopt_node_types_preserved:
assumes "w ∈ adopt_node_locs parent owner_document document_ptr"
assumes "h ⊢ w →⇩h h'"
shows "type_wf h = type_wf h'"
using assms
using remove_child_types_preserved
using set_disconnected_nodes_types_preserved
unfolding adopt_node_locs_def
by (auto split: if_splits)
end
locale l_adopt_node = l_type_wf + l_known_ptrs + l_get_parent_defs + l_adopt_node_defs +
l_get_child_nodes_defs + l_get_owner_document_defs +
assumes adopt_node_writes:
"writes (adopt_node_locs |h ⊢ get_parent node|⇩r
|h ⊢ get_owner_document (cast node)|⇩r document_ptr) (adopt_node document_ptr node) h h'"
assumes adopt_node_pointers_preserved:
"w ∈ adopt_node_locs parent owner_document document_ptr
⟹ h ⊢ w →⇩h h' ⟹ object_ptr_kinds h = object_ptr_kinds h'"
assumes adopt_node_types_preserved:
"w ∈ adopt_node_locs parent owner_document document_ptr
⟹ h ⊢ w →⇩h h' ⟹ type_wf h = type_wf h'"
assumes adopt_node_child_in_heap:
"h ⊢ ok (adopt_node document_ptr child) ⟹ child |∈| node_ptr_kinds h"
assumes adopt_node_children_subset:
"h ⊢ adopt_node owner_document node →⇩h h' ⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h' ⊢ get_child_nodes ptr →⇩r children'
⟹ known_ptrs h ⟹ type_wf h ⟹ set children' ⊆ set children"
interpretation
i_adopt_node?: l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_owner_document get_parent get_parent_locs remove_child
remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes
set_child_nodes_locs remove
apply(unfold_locales)
by(auto simp add: adopt_node_def adopt_node_locs_def)
declare l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma adopt_node_is_l_adopt_node [instances]:
"l_adopt_node type_wf known_ptr known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes
get_owner_document"
using instances
by (simp add: l_adopt_node_axioms_def adopt_node_child_in_heap adopt_node_children_subset
adopt_node_pointers_preserved adopt_node_types_preserved adopt_node_writes
l_adopt_node_def)
subsubsection ‹insert\_before›
locale l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_parent_defs get_parent get_parent_locs
+ l_get_child_nodes_defs get_child_nodes get_child_nodes_locs
+ l_set_child_nodes_defs set_child_nodes set_child_nodes_locs
+ l_get_ancestors_defs get_ancestors get_ancestors_locs
+ l_adopt_node_defs adopt_node adopt_node_locs
+ l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs
+ l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs
+ l_get_owner_document_defs get_owner_document
for get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_::linorder) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and adopt_node :: "(_) document_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and adopt_node_locs :: "(_) object_ptr option ⇒ (_) document_ptr ⇒ (_) document_ptr
⇒ ((_) heap, exception, unit) prog set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_owner_document :: "(_) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog"
begin
definition a_next_sibling :: "(_) node_ptr ⇒ (_, (_) node_ptr option) dom_prog"
where
"a_next_sibling node_ptr = do {
parent_opt ← get_parent node_ptr;
(case parent_opt of
Some parent ⇒ do {
children ← get_child_nodes parent;
(case (dropWhile (λptr. ptr = node_ptr) (dropWhile (λptr. ptr ≠ node_ptr) children)) of
x#_ ⇒ return (Some x)
| [] ⇒ return None)}
| None ⇒ return None)
}"
fun insert_before_list :: "'xyz ⇒ 'xyz option ⇒ 'xyz list ⇒ 'xyz list"
where
"insert_before_list v (Some reference) (x#xs) = (if reference = x
then v#x#xs else x # insert_before_list v (Some reference) xs)"
| "insert_before_list v (Some _) [] = [v]"
| "insert_before_list v None xs = xs @ [v]"
definition a_insert_node :: "(_) object_ptr ⇒ (_) node_ptr ⇒ (_) node_ptr option
⇒ (_, unit) dom_prog"
where
"a_insert_node ptr new_child reference_child_opt = do {
children ← get_child_nodes ptr;
set_child_nodes ptr (insert_before_list new_child reference_child_opt children)
}"
definition a_ensure_pre_insertion_validity :: "(_) node_ptr ⇒ (_) object_ptr
⇒ (_) node_ptr option ⇒ (_, unit) dom_prog"
where
"a_ensure_pre_insertion_validity node parent child_opt = do {
(if is_character_data_ptr_kind parent
then error HierarchyRequestError else return ());
ancestors ← get_ancestors parent;
(if cast node ∈ set ancestors then error HierarchyRequestError else return ());
(case child_opt of
Some child ⇒ do {
child_parent ← get_parent child;
(if child_parent ≠ Some parent then error NotFoundError else return ())}
| None ⇒ return ());
children ← get_child_nodes parent;
(if children ≠ [] ∧ is_document_ptr parent
then error HierarchyRequestError else return ());
(if is_character_data_ptr node ∧ is_document_ptr parent
then error HierarchyRequestError else return ())
}"
definition a_insert_before :: "(_) object_ptr ⇒ (_) node_ptr
⇒ (_) node_ptr option ⇒ (_, unit) dom_prog"
where
"a_insert_before ptr node child = do {
a_ensure_pre_insertion_validity node ptr child;
reference_child ← (if Some node = child
then a_next_sibling node
else return child);
owner_document ← get_owner_document ptr;
adopt_node owner_document node;
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (remove1 node disc_nodes);
a_insert_node ptr node reference_child
}"
definition a_insert_before_locs :: "(_) object_ptr ⇒ (_) object_ptr option ⇒ (_) document_ptr
⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
where
"a_insert_before_locs ptr old_parent child_owner_document ptr_owner_document =
adopt_node_locs old_parent child_owner_document ptr_owner_document ∪
set_child_nodes_locs ptr ∪
set_disconnected_nodes_locs ptr_owner_document"
end
locale l_insert_before_defs =
fixes insert_before :: "(_) object_ptr ⇒ (_) node_ptr ⇒ (_) node_ptr option ⇒ (_, unit) dom_prog"
fixes insert_before_locs :: "(_) object_ptr ⇒ (_) object_ptr option ⇒ (_) document_ptr
⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
locale l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_insert_before_defs
begin
definition "a_append_child ptr child = insert_before ptr child None"
end
locale l_append_child_defs =
fixes append_child :: "(_) object_ptr ⇒ (_) node_ptr ⇒ (_, unit) dom_prog"
locale l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes
set_child_nodes_locs get_ancestors get_ancestors_locs adopt_node adopt_node_locs
set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs get_owner_document
+ l_insert_before_defs
insert_before insert_before_locs
+ l_append_child_defs
append_child
+ l_set_child_nodes_get_child_nodes
type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs
+ l_get_ancestors
get_ancestors get_ancestors_locs
+ l_adopt_node
type_wf known_ptr known_ptrs get_parent get_parent_locs adopt_node adopt_node_locs
get_child_nodes get_child_nodes_locs get_owner_document
+ l_set_disconnected_nodes
type_wf set_disconnected_nodes set_disconnected_nodes_locs
+ l_get_disconnected_nodes
type_wf get_disconnected_nodes get_disconnected_nodes_locs
+ l_get_owner_document
get_owner_document
+ l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
+ l_set_disconnected_nodes_get_child_nodes
set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs
for get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_::linorder) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_child_nodes :: "(_) object_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and adopt_node :: "(_) document_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and adopt_node_locs :: "(_) object_ptr option ⇒ (_) document_ptr ⇒ (_) document_ptr
⇒ ((_) heap, exception, unit) prog set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_owner_document :: "(_) object_ptr ⇒ ((_) heap, exception, (_) document_ptr) prog"
and insert_before ::
"(_) object_ptr ⇒ (_) node_ptr ⇒ (_) node_ptr option ⇒ ((_) heap, exception, unit) prog"
and insert_before_locs :: "(_) object_ptr ⇒ (_) object_ptr option ⇒ (_) document_ptr
⇒ (_) document_ptr ⇒ (_, unit) dom_prog set"
and append_child :: "(_) object_ptr ⇒ (_) node_ptr ⇒ ((_) heap, exception, unit) prog"
and type_wf :: "(_) heap ⇒ bool"
and known_ptr :: "(_) object_ptr ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool" +
assumes insert_before_impl: "insert_before = a_insert_before"
assumes insert_before_locs_impl: "insert_before_locs = a_insert_before_locs"
begin
lemmas insert_before_def = a_insert_before_def[folded insert_before_impl]
lemmas insert_before_locs_def = a_insert_before_locs_def[folded insert_before_locs_impl]
lemma next_sibling_pure [simp]:
"pure (a_next_sibling new_child) h"
by(auto simp add: a_next_sibling_def get_parent_pure intro!: bind_pure_I split: option.splits list.splits)
lemma insert_before_list_in_set: "x ∈ set (insert_before_list v ref xs) ⟷ x = v ∨ x ∈ set xs"
apply(induct v ref xs rule: insert_before_list.induct)
by(auto)
lemma insert_before_list_distinct: "x ∉ set xs ⟹ distinct xs ⟹ distinct (insert_before_list x ref xs)"
apply(induct x ref xs rule: insert_before_list.induct)
by(auto simp add: insert_before_list_in_set)
lemma insert_before_list_subset: "set xs ⊆ set (insert_before_list x ref xs)"
apply(induct x ref xs rule: insert_before_list.induct)
by(auto)
lemma insert_before_list_node_in_set: "x ∈ set (insert_before_list x ref xs)"
apply(induct x ref xs rule: insert_before_list.induct)
by(auto)
lemma insert_node_writes:
"writes (set_child_nodes_locs ptr) (a_insert_node ptr new_child reference_child_opt) h h'"
by(auto simp add: a_insert_node_def set_child_nodes_writes
intro!: writes_bind_pure[OF get_child_nodes_pure])
lemma ensure_pre_insertion_validity_pure [simp]:
"pure (a_ensure_pre_insertion_validity node ptr child) h"
by(auto simp add: a_ensure_pre_insertion_validity_def
intro!: bind_pure_I
split: option.splits)
lemma insert_before_reference_child_not_in_children:
assumes "h ⊢ get_parent child →⇩r Some parent"
and "ptr ≠ parent"
and "¬is_character_data_ptr_kind ptr"
and "h ⊢ get_ancestors ptr →⇩r ancestors"
and "cast node ∉ set ancestors"
shows "h ⊢ insert_before ptr node (Some child) →⇩e NotFoundError"
proof -
have "h ⊢ a_ensure_pre_insertion_validity node ptr (Some child) →⇩e NotFoundError"
using assms unfolding insert_before_def a_ensure_pre_insertion_validity_def
by auto (simp | rule bind_returns_error_I2)+
then show ?thesis
unfolding insert_before_def by auto
qed
lemma insert_before_ptr_in_heap:
assumes "h ⊢ ok (insert_before ptr node reference_child)"
shows "ptr |∈| object_ptr_kinds h"
using assms
apply(auto simp add: insert_before_def elim!: bind_is_OK_E)[1]
by (metis (mono_tags, lifting) ensure_pre_insertion_validity_pure is_OK_returns_result_I
local.get_owner_document_ptr_in_heap next_sibling_pure pure_returns_heap_eq return_returns_heap)
lemma insert_before_child_in_heap:
assumes "h ⊢ ok (insert_before ptr node reference_child)"
shows "node |∈| node_ptr_kinds h"
using assms
apply(auto simp add: insert_before_def elim!: bind_is_OK_E)[1]
by (metis (mono_tags, lifting) ensure_pre_insertion_validity_pure is_OK_returns_heap_I
l_get_owner_document.get_owner_document_pure local.adopt_node_child_in_heap
local.l_get_owner_document_axioms next_sibling_pure pure_returns_heap_eq return_pure)
lemma insert_node_children_remain_distinct:
assumes insert_node: "h ⊢ a_insert_node ptr new_child reference_child_opt →⇩h h2"
and "h ⊢ get_child_nodes ptr →⇩r children"
and "new_child ∉ set children"
and "⋀ptr children. h ⊢ get_child_nodes ptr →⇩r children ⟹ distinct children"
and known_ptr: "known_ptr ptr"
and type_wf: "type_wf h"
shows "⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children ⟹ distinct children"
proof -
fix ptr' children'
assume a1: "h2 ⊢ get_child_nodes ptr' →⇩r children'"
then show "distinct children'"
proof (cases "ptr = ptr'")
case True
have "h2 ⊢ get_child_nodes ptr →⇩r (insert_before_list new_child reference_child_opt children)"
using assms(1) assms(2) apply(auto simp add: a_insert_node_def elim!: bind_returns_heap_E)[1]
using returns_result_eq set_child_nodes_get_child_nodes known_ptr type_wf
using pure_returns_heap_eq by fastforce
then show ?thesis
using True a1 assms(2) assms(3) assms(4) insert_before_list_distinct returns_result_eq
by fastforce
next
case False
have "h ⊢ get_child_nodes ptr' →⇩r children'"
using get_child_nodes_reads insert_node_writes insert_node a1
apply(rule reads_writes_separate_backwards)
by (meson False set_child_nodes_get_child_nodes_different_pointers)
then show ?thesis
using assms(4) by blast
qed
qed
lemma insert_before_writes:
"writes (insert_before_locs ptr |h ⊢ get_parent child|⇩r
|h ⊢ get_owner_document (cast child)|⇩r |h ⊢ get_owner_document ptr|⇩r) (insert_before ptr child ref) h h'"
apply(auto simp add: insert_before_def insert_before_locs_def a_insert_node_def
intro!: writes_bind)[1]
apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure local.adopt_node_writes
local.get_owner_document_pure next_sibling_pure pure_returns_heap_eq
select_result_I2 sup_commute writes_union_right_I)
apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure next_sibling_pure
pure_returns_heap_eq select_result_I2 set_disconnected_nodes_writes
writes_union_right_I)
apply (simp add: set_child_nodes_writes writes_union_left_I writes_union_right_I)
apply (metis (no_types, hide_lams) adopt_node_writes ensure_pre_insertion_validity_pure
get_owner_document_pure pure_returns_heap_eq select_result_I2 writes_union_left_I)
apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure pure_returns_heap_eq
select_result_I2 set_disconnected_nodes_writes writes_union_right_I)
by (simp add: set_child_nodes_writes writes_union_left_I writes_union_right_I)
end
locale l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_append_child_defs +
l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs +
assumes append_child_impl: "append_child = a_append_child"
begin
lemmas append_child_def = a_append_child_def[folded append_child_impl]
end
locale l_insert_before = l_insert_before_defs
locale l_append_child = l_append_child_defs
global_interpretation l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_parent get_parent_locs get_child_nodes
get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors get_ancestors_locs
adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs get_owner_document
defines
next_sibling = "l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_next_sibling get_parent get_child_nodes" and
insert_node = "l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_insert_node get_child_nodes set_child_nodes" and
ensure_pre_insertion_validity = "l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_ensure_pre_insertion_validity
get_parent get_child_nodes get_ancestors" and
insert_before = "l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_insert_before get_parent get_child_nodes
set_child_nodes get_ancestors adopt_node set_disconnected_nodes
get_disconnected_nodes get_owner_document" and
insert_before_locs = "l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_insert_before_locs set_child_nodes_locs
adopt_node_locs set_disconnected_nodes_locs"
.
global_interpretation l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs insert_before
defines append_child = "l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_append_child insert_before"
.
interpretation
i_insert_before?: l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_parent get_parent_locs get_child_nodes
get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors get_ancestors_locs
adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs append_child
type_wf known_ptr known_ptrs
apply(simp add: l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def instances)
by (simp add: insert_before_def insert_before_locs_def)
declare l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
interpretation i_append_child?: l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M append_child insert_before insert_before_locs
apply(simp add: l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances append_child_def)
done
declare l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
subsubsection ‹create\_element›
locale l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs +
l_set_tag_name_defs set_tag_name set_tag_name_locs
for get_disconnected_nodes ::
"(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs ::
"(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes ::
"(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs ::
"(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_tag_name ::
"(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_tag_name_locs ::
"(_) element_ptr ⇒ ((_) heap, exception, unit) prog set"
begin
definition a_create_element :: "(_) document_ptr ⇒ tag_name ⇒ (_, (_) element_ptr) dom_prog"
where
"a_create_element document_ptr tag = do {
new_element_ptr ← new_element;
set_tag_name new_element_ptr tag;
disc_nodes ← get_disconnected_nodes document_ptr;
set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes);
return new_element_ptr
}"
end
locale l_create_element_defs =
fixes create_element :: "(_) document_ptr ⇒ tag_name ⇒ (_, (_) element_ptr) dom_prog"
global_interpretation l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs
set_tag_name set_tag_name_locs
defines
create_element = "l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_create_element get_disconnected_nodes
set_disconnected_nodes set_tag_name"
.
locale l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs set_tag_name set_tag_name_locs +
l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs +
l_set_tag_name type_wf set_tag_name set_tag_name_locs +
l_create_element_defs create_element +
l_known_ptr known_ptr
for get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_tag_name :: "(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap, exception, unit) prog set"
and type_wf :: "(_) heap ⇒ bool"
and create_element :: "(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr) prog"
and known_ptr :: "(_) object_ptr ⇒ bool" +
assumes known_ptr_impl: "known_ptr = a_known_ptr"
assumes create_element_impl: "create_element = a_create_element"
begin
lemmas create_element_def = a_create_element_def[folded create_element_impl]
lemma create_element_document_in_heap:
assumes "h ⊢ ok (create_element document_ptr tag)"
shows "document_ptr |∈| document_ptr_kinds h"
proof -
obtain h' where "h ⊢ create_element document_ptr tag →⇩h h'"
using assms(1)
by auto
then
obtain new_element_ptr h2 h3 disc_nodes_h3 where
new_element_ptr: "h ⊢ new_element →⇩r new_element_ptr" and
h2: "h ⊢ new_element →⇩h h2" and
h3: "h2 ⊢ set_tag_name new_element_ptr tag →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes_h3) →⇩h h'"
by(auto simp add: create_element_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
have object_ptr_kinds_eq_h: "object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_element_ptr|}"
using new_element_new_ptr h2 new_element_ptr by blast
moreover have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_tag_name_writes h3])
using set_tag_name_pointers_preserved
by (auto simp add: reflp_def transp_def)
moreover have "document_ptr |∈| document_ptr_kinds h3"
by (meson disc_nodes_h3 is_OK_returns_result_I local.get_disconnected_nodes_ptr_in_heap)
ultimately show ?thesis
by (auto simp add: document_ptr_kinds_def)
qed
lemma create_element_known_ptr:
assumes "h ⊢ create_element document_ptr tag →⇩r new_element_ptr"
shows "known_ptr (cast new_element_ptr)"
proof -
have "is_element_ptr new_element_ptr"
using assms
apply(auto simp add: create_element_def elim!: bind_returns_result_E)[1]
using new_element_is_element_ptr
by blast
then show ?thesis
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs)
qed
end
locale l_create_element = l_create_element_defs
interpretation
i_create_element?: l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs set_tag_name set_tag_name_locs type_wf
create_element known_ptr
by(auto simp add: l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def create_element_def instances)
declare l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
subsubsection ‹create\_character\_data›
locale l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_set_val_defs set_val set_val_locs +
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs
for set_val :: "(_) character_data_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_val_locs :: "(_) character_data_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
begin
definition a_create_character_data :: "(_) document_ptr ⇒ string ⇒ (_, (_) character_data_ptr) dom_prog"
where
"a_create_character_data document_ptr text = do {
new_character_data_ptr ← new_character_data;
set_val new_character_data_ptr text;
disc_nodes ← get_disconnected_nodes document_ptr;
set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes);
return new_character_data_ptr
}"
end
locale l_create_character_data_defs =
fixes create_character_data :: "(_) document_ptr ⇒ string ⇒ (_, (_) character_data_ptr) dom_prog"
global_interpretation l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs set_val set_val_locs get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs
defines create_character_data = "l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_create_character_data
set_val get_disconnected_nodes set_disconnected_nodes"
.
locale l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs set_val set_val_locs get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs +
l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs +
l_set_val type_wf set_val set_val_locs +
l_create_character_data_defs create_character_data +
l_known_ptr known_ptr
for get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_val :: "(_) character_data_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_val_locs :: "(_) character_data_ptr ⇒ ((_) heap, exception, unit) prog set"
and type_wf :: "(_) heap ⇒ bool"
and create_character_data :: "(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) character_data_ptr) prog"
and known_ptr :: "(_) object_ptr ⇒ bool" +
assumes known_ptr_impl: "known_ptr = a_known_ptr"
assumes create_character_data_impl: "create_character_data = a_create_character_data"
begin
lemmas create_character_data_def = a_create_character_data_def[folded create_character_data_impl]
lemma create_character_data_document_in_heap:
assumes "h ⊢ ok (create_character_data document_ptr text)"
shows "document_ptr |∈| document_ptr_kinds h"
proof -
obtain h' where "h ⊢ create_character_data document_ptr text →⇩h h'"
using assms(1)
by auto
then
obtain new_character_data_ptr h2 h3 disc_nodes_h3 where
new_character_data_ptr: "h ⊢ new_character_data →⇩r new_character_data_ptr" and
h2: "h ⊢ new_character_data →⇩h h2" and
h3: "h2 ⊢ set_val new_character_data_ptr text →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes_h3) →⇩h h'"
by(auto simp add: create_character_data_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
have object_ptr_kinds_eq_h: "object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using new_character_data_new_ptr h2 new_character_data_ptr by blast
moreover have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_val_writes h3])
using set_val_pointers_preserved
by (auto simp add: reflp_def transp_def)
moreover have "document_ptr |∈| document_ptr_kinds h3"
by (meson disc_nodes_h3 is_OK_returns_result_I local.get_disconnected_nodes_ptr_in_heap)
ultimately show ?thesis
by (auto simp add: document_ptr_kinds_def)
qed
lemma create_character_data_known_ptr:
assumes "h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr"
shows "known_ptr (cast new_character_data_ptr)"
proof -
have "is_character_data_ptr new_character_data_ptr"
using assms
apply(auto simp add: create_character_data_def elim!: bind_returns_result_E)[1]
using new_character_data_is_character_data_ptr
by blast
then show ?thesis
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs)
qed
end
locale l_create_character_data = l_create_character_data_defs
interpretation
i_create_character_data?: l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs set_val set_val_locs
type_wf create_character_data known_ptr
by(auto simp add: l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def
create_character_data_def instances)
declare l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹create\_character\_data›
locale l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
begin
definition a_create_document :: "(_, (_) document_ptr) dom_prog"
where
"a_create_document = new_document"
end
locale l_create_document_defs =
fixes create_document :: "(_, (_) document_ptr) dom_prog"
global_interpretation l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs
defines create_document = "l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_create_document"
.
locale l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs +
l_create_document_defs +
assumes create_document_impl: "create_document = a_create_document"
begin
lemmas
create_document_def = create_document_impl[unfolded create_document_def, unfolded a_create_document_def]
end
locale l_create_document = l_create_document_defs
interpretation
i_create_document?: l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M create_document
by(simp add: l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹tree\_order›
locale l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_child_nodes_defs get_child_nodes get_child_nodes_locs
for get_child_nodes :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
partial_function (dom_prog) a_to_tree_order :: "(_) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
where
"a_to_tree_order ptr = (do {
children ← get_child_nodes ptr;
treeorders ← map_M a_to_tree_order (map (cast) children);
return (ptr # concat treeorders)
})"
end
locale l_to_tree_order_defs =
fixes to_tree_order :: "(_) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
global_interpretation l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs defines
to_tree_order = "l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_to_tree_order get_child_nodes" .
declare a_to_tree_order.simps [code]
locale l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs +
l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs +
l_to_tree_order_defs to_tree_order
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog" +
assumes to_tree_order_impl: "to_tree_order = a_to_tree_order"
begin
lemmas to_tree_order_def = a_to_tree_order.simps[folded to_tree_order_impl]
lemma to_tree_order_pure [simp]: "pure (to_tree_order ptr) h"
proof -
have "∀ptr h h' x. h ⊢ to_tree_order ptr →⇩r x ⟶ h ⊢ to_tree_order ptr →⇩h h' ⟶ h = h'"
proof (induct rule: a_to_tree_order.fixp_induct[folded to_tree_order_impl])
case 1
then show ?case
by (rule admissible_dom_prog)
next
case 2
then show ?case
by simp
next
case (3 f)
then have "⋀x h. pure (f x) h"
by (metis is_OK_returns_heap_E is_OK_returns_result_E pure_def)
then have "⋀xs h. pure (map_M f xs) h"
by(rule map_M_pure_I)
then show ?case
by(auto elim!: bind_returns_heap_E2)
qed
then show ?thesis
unfolding pure_def
by (metis is_OK_returns_heap_E is_OK_returns_result_E)
qed
end
locale l_to_tree_order =
fixes to_tree_order :: "(_) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
assumes to_tree_order_pure [simp]: "pure (to_tree_order ptr) h"
interpretation
i_to_tree_order?: l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes get_child_nodes_locs
to_tree_order
apply(unfold_locales)
by (simp add: to_tree_order_def)
declare l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma to_tree_order_is_l_to_tree_order [instances]: "l_to_tree_order to_tree_order"
using to_tree_order_pure l_to_tree_order_def by blast
subsubsection ‹first\_in\_tree\_order›
locale l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_to_tree_order_defs to_tree_order
for to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
begin
definition a_first_in_tree_order :: "(_) object_ptr ⇒ ((_) object_ptr
⇒ (_, 'result option) dom_prog) ⇒ (_, 'result option) dom_prog"
where
"a_first_in_tree_order ptr f = (do {
tree_order ← to_tree_order ptr;
results ← map_filter_M f tree_order;
(case results of
[] ⇒ return None
| x#_⇒ return (Some x))
})"
end
locale l_first_in_tree_order_defs =
fixes first_in_tree_order :: "(_) object_ptr ⇒ ((_) object_ptr ⇒ (_, 'result option) dom_prog)
⇒ (_, 'result option) dom_prog"
global_interpretation l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs to_tree_order defines
first_in_tree_order = "l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_first_in_tree_order to_tree_order" .
locale l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs to_tree_order +
l_first_in_tree_order_defs first_in_tree_order
for to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and first_in_tree_order :: "(_) object_ptr ⇒ ((_) object_ptr ⇒ ((_) heap, exception, 'result option) prog)
⇒ ((_) heap, exception, 'result option) prog" +
assumes first_in_tree_order_impl: "first_in_tree_order = a_first_in_tree_order"
begin
lemmas first_in_tree_order_def = first_in_tree_order_impl[unfolded a_first_in_tree_order_def]
end
locale l_first_in_tree_order
interpretation i_first_in_tree_order?:
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M to_tree_order first_in_tree_order
by unfold_locales (simp add: first_in_tree_order_def)
declare l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
subsubsection ‹get\_element\_by›
locale l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_first_in_tree_order_defs first_in_tree_order +
l_to_tree_order_defs to_tree_order +
l_get_attribute_defs get_attribute get_attribute_locs
for to_tree_order :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and first_in_tree_order :: "(_) object_ptr ⇒ ((_) object_ptr
⇒ ((_) heap, exception, (_) element_ptr option) prog)
⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_attribute :: "(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, char list option) prog"
and get_attribute_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
definition a_get_element_by_id :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr option) dom_prog"
where
"a_get_element_by_id ptr iden = first_in_tree_order ptr (λptr. (case cast ptr of
Some element_ptr ⇒ do {
id_opt ← get_attribute element_ptr ''id'';
(if id_opt = Some iden then return (Some element_ptr) else return None)
}
| _ ⇒ return None
))"
definition a_get_elements_by_class_name :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr list) dom_prog"
where
"a_get_elements_by_class_name ptr class_name = to_tree_order ptr ⤜
map_filter_M (λptr. (case cast ptr of
Some element_ptr ⇒ do {
class_name_opt ← get_attribute element_ptr ''class'';
(if class_name_opt = Some class_name then return (Some element_ptr) else return None)
}
| _ ⇒ return None))"
definition a_get_elements_by_tag_name :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr list) dom_prog"
where
"a_get_elements_by_tag_name ptr tag = to_tree_order ptr ⤜
map_filter_M (λptr. (case cast ptr of
Some element_ptr ⇒ do {
this_tag_name ← get_M element_ptr tag_name;
(if this_tag_name = tag then return (Some element_ptr) else return None)
}
| _ ⇒ return None))"
end
locale l_get_element_by_defs =
fixes get_element_by_id :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr option) dom_prog"
fixes get_elements_by_class_name :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr list) dom_prog"
fixes get_elements_by_tag_name :: "(_) object_ptr ⇒ attr_value ⇒ (_, (_) element_ptr list) dom_prog"
global_interpretation
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs to_tree_order first_in_tree_order get_attribute get_attribute_locs
defines
get_element_by_id = "l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_element_by_id first_in_tree_order get_attribute"
and
get_elements_by_class_name = "l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_elements_by_class_name
to_tree_order get_attribute"
and
get_elements_by_tag_name = "l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_get_elements_by_tag_name to_tree_order" .
locale l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs to_tree_order first_in_tree_order get_attribute get_attribute_locs +
l_get_element_by_defs get_element_by_id get_elements_by_class_name get_elements_by_tag_name +
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M to_tree_order first_in_tree_order +
l_to_tree_order to_tree_order +
l_get_attribute type_wf get_attribute get_attribute_locs
for to_tree_order :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and first_in_tree_order ::
"(_) object_ptr ⇒ ((_) object_ptr ⇒ ((_) heap, exception, (_) element_ptr option) prog)
⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_attribute :: "(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, char list option) prog"
and get_attribute_locs :: "(_) element_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_element_by_id ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_elements_by_class_name ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and get_elements_by_tag_name ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and type_wf :: "(_) heap ⇒ bool" +
assumes get_element_by_id_impl: "get_element_by_id = a_get_element_by_id"
assumes get_elements_by_class_name_impl: "get_elements_by_class_name = a_get_elements_by_class_name"
assumes get_elements_by_tag_name_impl: "get_elements_by_tag_name = a_get_elements_by_tag_name"
begin
lemmas
get_element_by_id_def = get_element_by_id_impl[unfolded a_get_element_by_id_def]
lemmas
get_elements_by_class_name_def = get_elements_by_class_name_impl[unfolded a_get_elements_by_class_name_def]
lemmas
get_elements_by_tag_name_def = get_elements_by_tag_name_impl[unfolded a_get_elements_by_tag_name_def]
lemma get_element_by_id_result_in_tree_order:
assumes "h ⊢ get_element_by_id ptr iden →⇩r Some element_ptr"
assumes "h ⊢ to_tree_order ptr →⇩r to"
shows "cast element_ptr ∈ set to"
using assms
by(auto simp add: get_element_by_id_def first_in_tree_order_def
elim!: map_filter_M_pure_E[where y=element_ptr] bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF assms(2), rotated]
intro!: map_filter_M_pure map_M_pure_I bind_pure_I
split: option.splits list.splits if_splits)
lemma get_elements_by_class_name_result_in_tree_order:
assumes "h ⊢ get_elements_by_class_name ptr name →⇩r results"
assumes "h ⊢ to_tree_order ptr →⇩r to"
assumes "element_ptr ∈ set results"
shows "cast element_ptr ∈ set to"
using assms
by(auto simp add: get_elements_by_class_name_def first_in_tree_order_def
elim!: map_filter_M_pure_E[where y=element_ptr] bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF assms(2), rotated]
intro!: map_filter_M_pure map_M_pure_I bind_pure_I
split: option.splits list.splits if_splits)
lemma get_elements_by_tag_name_result_in_tree_order:
assumes "h ⊢ get_elements_by_tag_name ptr name →⇩r results"
assumes "h ⊢ to_tree_order ptr →⇩r to"
assumes "element_ptr ∈ set results"
shows "cast element_ptr ∈ set to"
using assms
by(auto simp add: get_elements_by_tag_name_def first_in_tree_order_def
elim!: map_filter_M_pure_E[where y=element_ptr] bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF assms(2), rotated]
intro!: map_filter_M_pure map_M_pure_I bind_pure_I
split: option.splits list.splits if_splits)
lemma get_elements_by_tag_name_pure [simp]: "pure (get_elements_by_tag_name ptr tag) h"
by(auto simp add: get_elements_by_tag_name_def
intro!: bind_pure_I map_filter_M_pure
split: option.splits)
end
locale l_get_element_by = l_get_element_by_defs + l_to_tree_order_defs +
assumes get_element_by_id_result_in_tree_order:
"h ⊢ get_element_by_id ptr iden →⇩r Some element_ptr ⟹ h ⊢ to_tree_order ptr →⇩r to
⟹ cast element_ptr ∈ set to"
assumes get_elements_by_tag_name_pure [simp]: "pure (get_elements_by_tag_name ptr tag) h"
interpretation
i_get_element_by?: l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M to_tree_order first_in_tree_order get_attribute
get_attribute_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name type_wf
using instances
apply(simp add: l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def)
by(simp add: get_element_by_id_def get_elements_by_class_name_def get_elements_by_tag_name_def)
declare l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_element_by_is_l_get_element_by [instances]:
"l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order"
apply(unfold_locales)
using get_element_by_id_result_in_tree_order get_elements_by_tag_name_pure
by fast+
end
Theory Core_DOM_Heap_WF
section‹Wellformedness›
text‹In this theory, we discuss the wellformedness of the DOM. First, we define
wellformedness and, second, we show for all functions for querying and modifying the
DOM to what extend they preserve wellformendess.›
theory Core_DOM_Heap_WF
imports
"Core_DOM_Functions"
begin
locale l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs
for get_child_nodes :: "(_::linorder) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
definition a_owner_document_valid :: "(_) heap ⇒ bool"
where
"a_owner_document_valid h ⟷ (∀node_ptr ∈ fset (node_ptr_kinds h).
((∃document_ptr. document_ptr |∈| document_ptr_kinds h
∧ node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r)
∨ (∃parent_ptr. parent_ptr |∈| object_ptr_kinds h
∧ node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)))"
lemma a_owner_document_valid_code [code]: "a_owner_document_valid h ⟷ node_ptr_kinds h |⊆|
fset_of_list (concat (map (λparent. |h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h)) @ map (λparent. |h ⊢ get_disconnected_nodes parent|⇩r)
(sorted_list_of_fset (document_ptr_kinds h))))
"
apply(auto simp add: a_owner_document_valid_def
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_owner_document_valid_def)[1]
proof -
fix x
assume 1: " ∀node_ptr∈fset (node_ptr_kinds h).
(∃document_ptr. document_ptr |∈| document_ptr_kinds h ∧
node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r) ∨
(∃parent_ptr. parent_ptr |∈| object_ptr_kinds h ∧
node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)"
assume 2: "x |∈| node_ptr_kinds h"
assume 3: "x |∉| fset_of_list (concat (map (λparent. |h ⊢ get_disconnected_nodes parent|⇩r)
(sorted_list_of_fset (document_ptr_kinds h))))"
have "¬(∃document_ptr. document_ptr |∈| document_ptr_kinds h ∧
x ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r)"
using 1 2 3
by (smt UN_I fset_of_list_elem image_eqI notin_fset set_concat set_map sorted_list_of_fset_simps(1))
then
have "(∃parent_ptr. parent_ptr |∈| object_ptr_kinds h ∧ x ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)"
using 1 2
by auto
then obtain parent_ptr where parent_ptr:
"parent_ptr |∈| object_ptr_kinds h ∧ x ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r"
by auto
moreover have "parent_ptr ∈ set (sorted_list_of_fset (object_ptr_kinds h))"
using parent_ptr by auto
moreover have "|h ⊢ get_child_nodes parent_ptr|⇩r ∈ set (map (λparent. |h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h)))"
using calculation(2) by auto
ultimately
show "x |∈| fset_of_list (concat (map (λparent. |h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h))))"
using fset_of_list_elem by fastforce
next
fix node_ptr
assume 1: "node_ptr_kinds h |⊆| fset_of_list (concat (map (λparent. |h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h)))) |∪|
fset_of_list (concat (map (λparent. |h ⊢ get_disconnected_nodes parent|⇩r)
(sorted_list_of_fset (document_ptr_kinds h))))"
assume 2: "node_ptr |∈| node_ptr_kinds h"
assume 3: "∀parent_ptr. parent_ptr |∈| object_ptr_kinds h ⟶
node_ptr ∉ set |h ⊢ get_child_nodes parent_ptr|⇩r"
have "node_ptr ∈ set (concat (map (λparent. |h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h)))) ∨
node_ptr ∈ set (concat (map (λparent. |h ⊢ get_disconnected_nodes parent|⇩r)
(sorted_list_of_fset (document_ptr_kinds h))))"
using 1 2
by (meson fin_mono fset_of_list_elem funion_iff)
then
show "∃document_ptr. document_ptr |∈| document_ptr_kinds h ∧
node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r"
using 3
by auto
qed
definition a_parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
where
"a_parent_child_rel h = {(parent, child). parent |∈| object_ptr_kinds h
∧ child ∈ cast ` set |h ⊢ get_child_nodes parent|⇩r}"
lemma a_parent_child_rel_code [code]: "a_parent_child_rel h = set (concat (map
(λparent. map
(λchild. (parent, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child))
|h ⊢ get_child_nodes parent|⇩r)
(sorted_list_of_fset (object_ptr_kinds h)))
)"
by(auto simp add: a_parent_child_rel_def)
definition a_acyclic_heap :: "(_) heap ⇒ bool"
where
"a_acyclic_heap h = acyclic (a_parent_child_rel h)"
definition a_all_ptrs_in_heap :: "(_) heap ⇒ bool"
where
"a_all_ptrs_in_heap h ⟷
(∀ptr ∈ fset (object_ptr_kinds h). set |h ⊢ get_child_nodes ptr|⇩r ⊆ fset (node_ptr_kinds h)) ∧
(∀document_ptr ∈ fset (document_ptr_kinds h).
set |h ⊢ get_disconnected_nodes document_ptr|⇩r ⊆ fset (node_ptr_kinds h))"
definition a_distinct_lists :: "(_) heap ⇒ bool"
where
"a_distinct_lists h = distinct (concat (
(map (λptr. |h ⊢ get_child_nodes ptr|⇩r) |h ⊢ object_ptr_kinds_M|⇩r)
@ (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r) |h ⊢ document_ptr_kinds_M|⇩r)
))"
definition a_heap_is_wellformed :: "(_) heap ⇒ bool"
where
"a_heap_is_wellformed h ⟷
a_acyclic_heap h ∧ a_all_ptrs_in_heap h ∧ a_distinct_lists h ∧ a_owner_document_valid h"
end
locale l_heap_is_wellformed_defs =
fixes heap_is_wellformed :: "(_) heap ⇒ bool"
fixes parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
global_interpretation l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
defines heap_is_wellformed = "l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_heap_is_wellformed get_child_nodes
get_disconnected_nodes"
and parent_child_rel = "l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_parent_child_rel get_child_nodes"
and acyclic_heap = a_acyclic_heap
and all_ptrs_in_heap = a_all_ptrs_in_heap
and distinct_lists = a_distinct_lists
and owner_document_valid = a_owner_document_valid
.
locale l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs
+ l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_child_nodes get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs
+ l_heap_is_wellformed_defs heap_is_wellformed parent_child_rel
+ l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set" +
assumes heap_is_wellformed_impl: "heap_is_wellformed = a_heap_is_wellformed"
assumes parent_child_rel_impl: "parent_child_rel = a_parent_child_rel"
begin
lemmas heap_is_wellformed_def = heap_is_wellformed_impl[unfolded a_heap_is_wellformed_def]
lemmas parent_child_rel_def = parent_child_rel_impl[unfolded a_parent_child_rel_def]
lemmas acyclic_heap_def = a_acyclic_heap_def[folded parent_child_rel_impl]
lemma parent_child_rel_node_ptr:
"(parent, child) ∈ parent_child_rel h ⟹ is_node_ptr_kind child"
by(auto simp add: parent_child_rel_def)
lemma parent_child_rel_child_nodes:
assumes "known_ptr parent"
and "h ⊢ get_child_nodes parent →⇩r children"
and "child ∈ set children"
shows "(parent, cast child) ∈ parent_child_rel h"
using assms
apply(auto simp add: parent_child_rel_def is_OK_returns_result_I )[1]
using get_child_nodes_ptr_in_heap by blast
lemma parent_child_rel_child_nodes2:
assumes "known_ptr parent"
and "h ⊢ get_child_nodes parent →⇩r children"
and "child ∈ set children"
and "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child = child_obj"
shows "(parent, child_obj) ∈ parent_child_rel h"
using assms parent_child_rel_child_nodes by blast
lemma parent_child_rel_finite: "finite (parent_child_rel h)"
proof -
have "parent_child_rel h = (⋃ptr ∈ set |h ⊢ object_ptr_kinds_M|⇩r.
(⋃child ∈ set |h ⊢ get_child_nodes ptr|⇩r. {(ptr, cast child)}))"
by(auto simp add: parent_child_rel_def)
moreover have "finite (⋃ptr ∈ set |h ⊢ object_ptr_kinds_M|⇩r.
(⋃child ∈ set |h ⊢ get_child_nodes ptr|⇩r. {(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child)}))"
by simp
ultimately show ?thesis
by simp
qed
lemma distinct_lists_no_parent:
assumes "a_distinct_lists h"
assumes "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assumes "node_ptr ∈ set disc_nodes"
shows "¬(∃parent_ptr. parent_ptr |∈| object_ptr_kinds h
∧ node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)"
using assms
apply(auto simp add: a_distinct_lists_def)[1]
proof -
fix parent_ptr :: "(_) object_ptr"
assume a1: "parent_ptr |∈| object_ptr_kinds h"
assume a2: "(⋃x∈fset (object_ptr_kinds h).
set |h ⊢ get_child_nodes x|⇩r) ∩ (⋃x∈fset (document_ptr_kinds h).
set |h ⊢ get_disconnected_nodes x|⇩r) = {}"
assume a3: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assume a4: "node_ptr ∈ set disc_nodes"
assume a5: "node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r"
have f6: "parent_ptr ∈ fset (object_ptr_kinds h)"
using a1 by auto
have f7: "document_ptr ∈ fset (document_ptr_kinds h)"
using a3 by (meson fmember.rep_eq get_disconnected_nodes_ptr_in_heap is_OK_returns_result_I)
have "|h ⊢ get_disconnected_nodes document_ptr|⇩r = disc_nodes"
using a3 by simp
then show False
using f7 f6 a5 a4 a2 by blast
qed
lemma distinct_lists_disconnected_nodes:
assumes "a_distinct_lists h"
and "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
shows "distinct disc_nodes"
proof -
have h1: "distinct (concat (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r)
|h ⊢ document_ptr_kinds_M|⇩r))"
using assms(1)
by(simp add: a_distinct_lists_def)
then show ?thesis
using concat_map_all_distinct[OF h1] assms(2) is_OK_returns_result_I get_disconnected_nodes_ok
by (metis (no_types, lifting) DocumentMonad.ptr_kinds_ptr_kinds_M
l_get_disconnected_nodes.get_disconnected_nodes_ptr_in_heap
l_get_disconnected_nodes_axioms select_result_I2)
qed
lemma distinct_lists_children:
assumes "a_distinct_lists h"
and "known_ptr ptr"
and "h ⊢ get_child_nodes ptr →⇩r children"
shows "distinct children"
proof (cases "children = []", simp)
assume "children ≠ []"
have h1: "distinct (concat ((map (λptr. |h ⊢ get_child_nodes ptr|⇩r) |h ⊢ object_ptr_kinds_M|⇩r)))"
using assms(1)
by(simp add: a_distinct_lists_def)
show ?thesis
using concat_map_all_distinct[OF h1] assms(2) assms(3)
by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M get_child_nodes_ptr_in_heap
is_OK_returns_result_I select_result_I2)
qed
lemma heap_is_wellformed_children_in_heap:
assumes "heap_is_wellformed h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "child ∈ set children"
shows "child |∈| node_ptr_kinds h"
using assms
apply(auto simp add: heap_is_wellformed_def a_all_ptrs_in_heap_def)[1]
by (metis (no_types, lifting) finite_set_in is_OK_returns_result_I
local.get_child_nodes_ptr_in_heap select_result_I2 subsetD)
lemma heap_is_wellformed_one_parent:
assumes "heap_is_wellformed h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "h ⊢ get_child_nodes ptr' →⇩r children'"
assumes "set children ∩ set children' ≠ {}"
shows "ptr = ptr'"
using assms
proof (auto simp add: heap_is_wellformed_def a_distinct_lists_def)[1]
fix x :: "(_) node_ptr"
assume a1: "ptr ≠ ptr'"
assume a2: "h ⊢ get_child_nodes ptr →⇩r children"
assume a3: "h ⊢ get_child_nodes ptr' →⇩r children'"
assume a4: "distinct (concat (map (λptr. |h ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h)))))"
have f5: "|h ⊢ get_child_nodes ptr|⇩r = children"
using a2 by simp
have "|h ⊢ get_child_nodes ptr'|⇩r = children'"
using a3 by (meson select_result_I2)
then have "ptr ∉ set (sorted_list_of_set (fset (object_ptr_kinds h)))
∨ ptr' ∉ set (sorted_list_of_set (fset (object_ptr_kinds h)))
∨ set children ∩ set children' = {}"
using f5 a4 a1 by (meson distinct_concat_map_E(1))
then show False
using a3 a2 by (metis (no_types) assms(4) finite_fset fmember.rep_eq is_OK_returns_result_I
local.get_child_nodes_ptr_in_heap set_sorted_list_of_set)
qed
lemma parent_child_rel_child:
"h ⊢ get_child_nodes ptr →⇩r children ⟹
child ∈ set children ⟷ (ptr, cast child) ∈ parent_child_rel h"
by (simp add: is_OK_returns_result_I get_child_nodes_ptr_in_heap parent_child_rel_def)
lemma parent_child_rel_acyclic: "heap_is_wellformed h ⟹ acyclic (parent_child_rel h)"
by (simp add: acyclic_heap_def local.heap_is_wellformed_def)
lemma heap_is_wellformed_disconnected_nodes_distinct:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes ⟹
distinct disc_nodes"
using distinct_lists_disconnected_nodes local.heap_is_wellformed_def by blast
lemma parent_child_rel_parent_in_heap:
"(parent, child_ptr) ∈ parent_child_rel h ⟹ parent |∈| object_ptr_kinds h"
using local.parent_child_rel_def by blast
lemma parent_child_rel_child_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptr parent
⟹ (parent, child_ptr) ∈ parent_child_rel h ⟹ child_ptr |∈| object_ptr_kinds h"
apply(auto simp add: heap_is_wellformed_def parent_child_rel_def a_all_ptrs_in_heap_def)[1]
using get_child_nodes_ok
by (meson finite_set_in subsetD)
lemma heap_is_wellformed_disc_nodes_in_heap:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ node ∈ set disc_nodes ⟹ node |∈| node_ptr_kinds h"
by (metis (no_types, lifting) finite_set_in is_OK_returns_result_I local.a_all_ptrs_in_heap_def
local.get_disconnected_nodes_ptr_in_heap local.heap_is_wellformed_def select_result_I2 subsetD)
lemma heap_is_wellformed_one_disc_parent:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ h ⊢ get_disconnected_nodes document_ptr' →⇩r disc_nodes'
⟹ set disc_nodes ∩ set disc_nodes' ≠ {} ⟹ document_ptr = document_ptr'"
using DocumentMonad.ptr_kinds_ptr_kinds_M concat_append distinct_append distinct_concat_map_E(1)
is_OK_returns_result_I local.a_distinct_lists_def local.get_disconnected_nodes_ptr_in_heap
local.heap_is_wellformed_def select_result_I2
proof -
assume a1: "heap_is_wellformed h"
assume a2: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assume a3: "h ⊢ get_disconnected_nodes document_ptr' →⇩r disc_nodes'"
assume a4: "set disc_nodes ∩ set disc_nodes' ≠ {}"
have f5: "|h ⊢ get_disconnected_nodes document_ptr|⇩r = disc_nodes"
using a2 by (meson select_result_I2)
have f6: "|h ⊢ get_disconnected_nodes document_ptr'|⇩r = disc_nodes'"
using a3 by (meson select_result_I2)
have "⋀nss nssa. ¬ distinct (concat (nss @ nssa)) ∨ distinct (concat nssa::(_) node_ptr list)"
by (metis (no_types) concat_append distinct_append)
then have "distinct (concat (map (λd. |h ⊢ get_disconnected_nodes d|⇩r) |h ⊢ document_ptr_kinds_M|⇩r))"
using a1 local.a_distinct_lists_def local.heap_is_wellformed_def by blast
then show ?thesis
using f6 f5 a4 a3 a2 by (meson DocumentMonad.ptr_kinds_ptr_kinds_M distinct_concat_map_E(1)
is_OK_returns_result_I local.get_disconnected_nodes_ptr_in_heap)
qed
lemma heap_is_wellformed_children_disc_nodes_different:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ set children ∩ set disc_nodes = {}"
by (metis (no_types, hide_lams) disjoint_iff_not_equal distinct_lists_no_parent
is_OK_returns_result_I local.get_child_nodes_ptr_in_heap
local.heap_is_wellformed_def select_result_I2)
lemma heap_is_wellformed_children_disc_nodes:
"heap_is_wellformed h ⟹ node_ptr |∈| node_ptr_kinds h
⟹ ¬(∃parent ∈ fset (object_ptr_kinds h). node_ptr ∈ set |h ⊢ get_child_nodes parent|⇩r)
⟹ (∃document_ptr ∈ fset (document_ptr_kinds h). node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r)"
apply(auto simp add: heap_is_wellformed_def a_distinct_lists_def a_owner_document_valid_def)[1]
by (meson fmember.rep_eq)
lemma heap_is_wellformed_children_distinct:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children ⟹ distinct children"
by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M concat_append distinct_append
distinct_concat_map_E(2) is_OK_returns_result_I local.a_distinct_lists_def
local.get_child_nodes_ptr_in_heap local.heap_is_wellformed_def
select_result_I2)
end
locale l_heap_is_wellformed = l_type_wf + l_known_ptr + l_heap_is_wellformed_defs
+ l_get_child_nodes_defs + l_get_disconnected_nodes_defs +
assumes heap_is_wellformed_children_in_heap:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children ⟹ child ∈ set children
⟹ child |∈| node_ptr_kinds h"
assumes heap_is_wellformed_disc_nodes_in_heap:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ node ∈ set disc_nodes ⟹ node |∈| node_ptr_kinds h"
assumes heap_is_wellformed_one_parent:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h ⊢ get_child_nodes ptr' →⇩r children'
⟹ set children ∩ set children' ≠ {} ⟹ ptr = ptr'"
assumes heap_is_wellformed_one_disc_parent:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ h ⊢ get_disconnected_nodes document_ptr' →⇩r disc_nodes'
⟹ set disc_nodes ∩ set disc_nodes' ≠ {} ⟹ document_ptr = document_ptr'"
assumes heap_is_wellformed_children_disc_nodes_different:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ set children ∩ set disc_nodes = {}"
assumes heap_is_wellformed_disconnected_nodes_distinct:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes
⟹ distinct disc_nodes"
assumes heap_is_wellformed_children_distinct:
"heap_is_wellformed h ⟹ h ⊢ get_child_nodes ptr →⇩r children ⟹ distinct children"
assumes heap_is_wellformed_children_disc_nodes:
"heap_is_wellformed h ⟹ node_ptr |∈| node_ptr_kinds h
⟹ ¬(∃parent ∈ fset (object_ptr_kinds h). node_ptr ∈ set |h ⊢ get_child_nodes parent|⇩r)
⟹ (∃document_ptr ∈ fset (document_ptr_kinds h). node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r)"
assumes parent_child_rel_child:
"h ⊢ get_child_nodes ptr →⇩r children
⟹ child ∈ set children ⟷ (ptr, cast child) ∈ parent_child_rel h"
assumes parent_child_rel_finite:
"heap_is_wellformed h ⟹ finite (parent_child_rel h)"
assumes parent_child_rel_acyclic:
"heap_is_wellformed h ⟹ acyclic (parent_child_rel h)"
assumes parent_child_rel_node_ptr:
"(parent, child_ptr) ∈ parent_child_rel h ⟹ is_node_ptr_kind child_ptr"
assumes parent_child_rel_parent_in_heap:
"(parent, child_ptr) ∈ parent_child_rel h ⟹ parent |∈| object_ptr_kinds h"
assumes parent_child_rel_child_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptr parent
⟹ (parent, child_ptr) ∈ parent_child_rel h ⟹ child_ptr |∈| object_ptr_kinds h"
interpretation i_heap_is_wellformed?: l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes
get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
heap_is_wellformed parent_child_rel
apply(unfold_locales)
by(auto simp add: heap_is_wellformed_def parent_child_rel_def)
declare l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma heap_is_wellformed_is_l_heap_is_wellformed [instances]:
"l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes
get_disconnected_nodes"
apply(auto simp add: l_heap_is_wellformed_def)[1]
using heap_is_wellformed_children_in_heap
apply blast
using heap_is_wellformed_disc_nodes_in_heap
apply blast
using heap_is_wellformed_one_parent
apply blast
using heap_is_wellformed_one_disc_parent
apply blast
using heap_is_wellformed_children_disc_nodes_different
apply blast
using heap_is_wellformed_disconnected_nodes_distinct
apply blast
using heap_is_wellformed_children_distinct
apply blast
using heap_is_wellformed_children_disc_nodes
apply blast
using parent_child_rel_child
apply (blast)
using parent_child_rel_child
apply(blast)
using parent_child_rel_finite
apply blast
using parent_child_rel_acyclic
apply blast
using parent_child_rel_node_ptr
apply blast
using parent_child_rel_parent_in_heap
apply blast
using parent_child_rel_child_in_heap
apply blast
done
subsection ‹get\_parent›
locale l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
+ l_heap_is_wellformed
type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and known_ptrs :: "(_) heap ⇒ bool"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma child_parent_dual:
assumes heap_is_wellformed: "heap_is_wellformed h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "child ∈ set children"
assumes "known_ptrs h"
assumes type_wf: "type_wf h"
shows "h ⊢ get_parent child →⇩r Some ptr"
proof -
obtain ptrs where ptrs: "h ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have h1: "ptr ∈ set ptrs"
using get_child_nodes_ok assms(2) is_OK_returns_result_I
by (metis (no_types, hide_lams) ObjectMonad.ptr_kinds_ptr_kinds_M
‹⋀thesis. (⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs ⟹ thesis) ⟹ thesis›
get_child_nodes_ptr_in_heap returns_result_eq select_result_I2)
let ?P = "(λptr. get_child_nodes ptr ⤜ (λchildren. return (child ∈ set children)))"
let ?filter = "filter_M ?P ptrs"
have "h ⊢ ok ?filter"
using ptrs type_wf
using get_child_nodes_ok
apply(auto intro!: filter_M_is_OK_I bind_is_OK_pure_I get_child_nodes_ok simp add: bind_pure_I)[1]
using assms(4) local.known_ptrs_known_ptr by blast
then obtain parent_ptrs where parent_ptrs: "h ⊢ ?filter →⇩r parent_ptrs"
by auto
have h5: "∃!x. x ∈ set ptrs ∧ h ⊢ Heap_Error_Monad.bind (get_child_nodes x)
(λchildren. return (child ∈ set children)) →⇩r True"
apply(auto intro!: bind_pure_returns_result_I)[1]
using heap_is_wellformed_one_parent
proof -
have "h ⊢ (return (child ∈ set children)::((_) heap, exception, bool) prog) →⇩r True"
by (simp add: assms(3))
then show
"∃z. z ∈ set ptrs ∧ h ⊢ Heap_Error_Monad.bind (get_child_nodes z)
(λns. return (child ∈ set ns)) →⇩r True"
by (metis (no_types) assms(2) bind_pure_returns_result_I2 h1 is_OK_returns_result_I
local.get_child_nodes_pure select_result_I2)
next
fix x y
assume 0: "x ∈ set ptrs"
and 1: "h ⊢ Heap_Error_Monad.bind (get_child_nodes x)
(λchildren. return (child ∈ set children)) →⇩r True"
and 2: "y ∈ set ptrs"
and 3: "h ⊢ Heap_Error_Monad.bind (get_child_nodes y)
(λchildren. return (child ∈ set children)) →⇩r True"
and 4: "(⋀h ptr children ptr' children'. heap_is_wellformed h
⟹ h ⊢ get_child_nodes ptr →⇩r children ⟹ h ⊢ get_child_nodes ptr' →⇩r children'
⟹ set children ∩ set children' ≠ {} ⟹ ptr = ptr')"
then show "x = y"
by (metis (no_types, lifting) bind_returns_result_E disjoint_iff_not_equal heap_is_wellformed
return_returns_result)
qed
have "child |∈| node_ptr_kinds h"
using heap_is_wellformed_children_in_heap heap_is_wellformed assms(2) assms(3)
by fast
moreover have "parent_ptrs = [ptr]"
apply(rule filter_M_ex1[OF parent_ptrs h1 h5])
using ptrs assms(2) assms(3)
by(auto simp add: object_ptr_kinds_M_defs bind_pure_I intro!: bind_pure_returns_result_I)
ultimately show ?thesis
using ptrs parent_ptrs
by(auto simp add: bind_pure_I get_parent_def
elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I filter_M_pure_I)
qed
lemma parent_child_rel_parent:
assumes "heap_is_wellformed h"
and "h ⊢ get_parent child_node →⇩r Some parent"
shows "(parent, cast child_node) ∈ parent_child_rel h"
using assms parent_child_rel_child get_parent_child_dual by auto
lemma heap_wellformed_induct [consumes 1, case_names step]:
assumes "heap_is_wellformed h"
and step: "⋀parent. (⋀children child. h ⊢ get_child_nodes parent →⇩r children
⟹ child ∈ set children ⟹ P (cast child)) ⟹ P parent"
shows "P ptr"
proof -
fix ptr
have "wf ((parent_child_rel h)¯)"
by (simp add: assms(1) finite_acyclic_wf_converse parent_child_rel_acyclic parent_child_rel_finite)
then show "?thesis"
proof (induct rule: wf_induct_rule)
case (less parent)
then show ?case
using assms parent_child_rel_child
by (meson converse_iff)
qed
qed
lemma heap_wellformed_induct2 [consumes 3, case_names not_in_heap empty_children step]:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
and not_in_heap: "⋀parent. parent |∉| object_ptr_kinds h ⟹ P parent"
and empty_children: "⋀parent. h ⊢ get_child_nodes parent →⇩r [] ⟹ P parent"
and step: "⋀parent children child. h ⊢ get_child_nodes parent →⇩r children
⟹ child ∈ set children ⟹ P (cast child) ⟹ P parent"
shows "P ptr"
proof(insert assms(1), induct rule: heap_wellformed_induct)
case (step parent)
then show ?case
proof(cases "parent |∈| object_ptr_kinds h")
case True
then obtain children where children: "h ⊢ get_child_nodes parent →⇩r children"
using get_child_nodes_ok assms(2) assms(3)
by (meson is_OK_returns_result_E local.known_ptrs_known_ptr)
then show ?thesis
proof (cases "children = []")
case True
then show ?thesis
using children empty_children
by simp
next
case False
then show ?thesis
using assms(6) children last_in_set step.hyps by blast
qed
next
case False
then show ?thesis
by (simp add: not_in_heap)
qed
qed
lemma heap_wellformed_induct_rev [consumes 1, case_names step]:
assumes "heap_is_wellformed h"
and step: "⋀child. (⋀parent child_node. cast child_node = child
⟹ h ⊢ get_parent child_node →⇩r Some parent ⟹ P parent) ⟹ P child"
shows "P ptr"
proof -
fix ptr
have "wf ((parent_child_rel h))"
by (simp add: assms(1) local.parent_child_rel_acyclic local.parent_child_rel_finite
wf_iff_acyclic_if_finite)
then show "?thesis"
proof (induct rule: wf_induct_rule)
case (less child)
show ?case
using assms get_parent_child_dual
by (metis less.hyps parent_child_rel_parent)
qed
qed
end
interpretation i_get_parent_wf?: l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes
get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed
parent_child_rel get_disconnected_nodes
using instances
by(simp add: l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
locale l_get_parent_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
heap_is_wellformed parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs
+ l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs heap_is_wellformed parent_child_rel
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and known_ptrs :: "(_) heap ⇒ bool"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma preserves_wellformedness_writes_needed:
assumes heap_is_wellformed: "heap_is_wellformed h"
and "h ⊢ f →⇩h h'"
and "writes SW f h h'"
and preserved_get_child_nodes:
"⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. ∀r ∈ get_child_nodes_locs object_ptr. r h h'"
and preserved_get_disconnected_nodes:
"⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀document_ptr. ∀r ∈ get_disconnected_nodes_locs document_ptr. r h h'"
and preserved_object_pointers:
"⋀h h' w. w ∈ SW ⟹ h ⊢ w →⇩h h'
⟹ ∀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
shows "heap_is_wellformed h'"
proof -
have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
using assms(2) assms(3) object_ptr_kinds_preserved preserved_object_pointers by blast
then have object_ptr_kinds_eq:
"⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
unfolding object_ptr_kinds_M_defs by simp
then have object_ptr_kinds_eq2: "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
using select_result_eq by force
then have node_ptr_kinds_eq2: "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by auto
then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'"
by auto
have document_ptr_kinds_eq2: "|h ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'"
by auto
have children_eq:
"⋀ptr children. h ⊢ get_child_nodes ptr →⇩r children = h' ⊢ get_child_nodes ptr →⇩r children"
apply(rule reads_writes_preserved[OF get_child_nodes_reads assms(3) assms(2)])
using preserved_get_child_nodes by fast
then have children_eq2: "⋀ptr. |h ⊢ get_child_nodes ptr|⇩r = |h' ⊢ get_child_nodes ptr|⇩r"
using select_result_eq by force
have disconnected_nodes_eq:
"⋀document_ptr disconnected_nodes.
h ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes
= h' ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes"
apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads assms(3) assms(2)])
using preserved_get_disconnected_nodes by fast
then have disconnected_nodes_eq2:
"⋀document_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r
= |h' ⊢ get_disconnected_nodes document_ptr|⇩r"
using select_result_eq by force
have get_parent_eq: "⋀ptr parent. h ⊢ get_parent ptr →⇩r parent = h' ⊢ get_parent ptr →⇩r parent"
apply(rule reads_writes_preserved[OF get_parent_reads assms(3) assms(2)])
using preserved_get_child_nodes preserved_object_pointers unfolding get_parent_locs_def by fast
have "a_acyclic_heap h"
using heap_is_wellformed by (simp add: heap_is_wellformed_def)
have "parent_child_rel h' ⊆ parent_child_rel h"
proof
fix x
assume "x ∈ parent_child_rel h'"
then show "x ∈ parent_child_rel h"
by(simp add: parent_child_rel_def children_eq2 object_ptr_kinds_eq3)
qed
then have "a_acyclic_heap h'"
using ‹a_acyclic_heap h› acyclic_heap_def acyclic_subset by blast
moreover have "a_all_ptrs_in_heap h"
using heap_is_wellformed by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h'"
by (simp add: children_eq2 disconnected_nodes_eq2 document_ptr_kinds_eq3
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.a_all_ptrs_in_heap_def node_ptr_kinds_eq3 object_ptr_kinds_eq3)
moreover have h0: "a_distinct_lists h"
using heap_is_wellformed by (simp add: heap_is_wellformed_def)
have h1: "map (λptr. |h ⊢ get_child_nodes ptr|⇩r) (sorted_list_of_set (fset (object_ptr_kinds h)))
= map (λptr. |h' ⊢ get_child_nodes ptr|⇩r) (sorted_list_of_set (fset (object_ptr_kinds h')))"
by (simp add: children_eq2 object_ptr_kinds_eq3)
have h2: "map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h)))
= map (λdocument_ptr. |h' ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h')))"
using disconnected_nodes_eq document_ptr_kinds_eq2 select_result_eq by force
have "a_distinct_lists h'"
using h0
by(simp add: a_distinct_lists_def h1 h2)
moreover have "a_owner_document_valid h"
using heap_is_wellformed by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
by(auto simp add: a_owner_document_valid_def children_eq2 disconnected_nodes_eq2
object_ptr_kinds_eq3 node_ptr_kinds_eq3 document_ptr_kinds_eq3)
ultimately show ?thesis
by (simp add: heap_is_wellformed_def)
qed
end
interpretation i_get_parent_wf2?: l_get_parent_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes
get_child_nodes_locs known_ptrs get_parent get_parent_locs
heap_is_wellformed parent_child_rel get_disconnected_nodes
get_disconnected_nodes_locs
using l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms
by (simp add: l_get_parent_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_get_parent_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
locale l_get_parent_wf = l_type_wf + l_known_ptrs + l_heap_is_wellformed_defs
+ l_get_child_nodes_defs + l_get_parent_defs +
assumes child_parent_dual:
"heap_is_wellformed h
⟹ type_wf h
⟹ known_ptrs h
⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ child ∈ set children
⟹ h ⊢ get_parent child →⇩r Some ptr"
assumes heap_wellformed_induct [consumes 1, case_names step]:
"heap_is_wellformed h
⟹ (⋀parent. (⋀children child. h ⊢ get_child_nodes parent →⇩r children
⟹ child ∈ set children ⟹ P (cast child)) ⟹ P parent)
⟹ P ptr"
assumes heap_wellformed_induct_rev [consumes 1, case_names step]:
"heap_is_wellformed h
⟹ (⋀child. (⋀parent child_node. cast child_node = child
⟹ h ⊢ get_parent child_node →⇩r Some parent ⟹ P parent) ⟹ P child)
⟹ P ptr"
assumes parent_child_rel_parent: "heap_is_wellformed h
⟹ h ⊢ get_parent child_node →⇩r Some parent
⟹ (parent, cast child_node) ∈ parent_child_rel h"
lemma get_parent_wf_is_l_get_parent_wf [instances]:
"l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel
get_child_nodes get_parent"
using known_ptrs_is_l_known_ptrs
apply(auto simp add: l_get_parent_wf_def l_get_parent_wf_axioms_def)[1]
using child_parent_dual heap_wellformed_induct heap_wellformed_induct_rev parent_child_rel_parent
by metis+
subsection ‹get\_disconnected\_nodes›
subsection ‹set\_disconnected\_nodes›
subsubsection ‹get\_disconnected\_nodes›
locale l_set_disconnected_nodes_get_disconnected_nodes_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_set_disconnected_nodes_get_disconnected_nodes
type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs
+ l_heap_is_wellformed
type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
for known_ptr :: "(_) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma remove_from_disconnected_nodes_removes:
assumes "heap_is_wellformed h"
assumes "h ⊢ get_disconnected_nodes ptr →⇩r disc_nodes"
assumes "h ⊢ set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) →⇩h h'"
assumes "h' ⊢ get_disconnected_nodes ptr →⇩r disc_nodes'"
shows "node_ptr ∉ set disc_nodes'"
using assms
by (metis distinct_remove1_removeAll heap_is_wellformed_disconnected_nodes_distinct
set_disconnected_nodes_get_disconnected_nodes member_remove remove_code(1)
returns_result_eq)
end
locale l_set_disconnected_nodes_get_disconnected_nodes_wf = l_heap_is_wellformed
+ l_set_disconnected_nodes_get_disconnected_nodes +
assumes remove_from_disconnected_nodes_removes:
"heap_is_wellformed h ⟹ h ⊢ get_disconnected_nodes ptr →⇩r disc_nodes
⟹ h ⊢ set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) →⇩h h'
⟹ h' ⊢ get_disconnected_nodes ptr →⇩r disc_nodes'
⟹ node_ptr ∉ set disc_nodes'"
interpretation i_set_disconnected_nodes_get_disconnected_nodes_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M?:
l_set_disconnected_nodes_get_disconnected_nodes_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed
parent_child_rel get_child_nodes
using instances
by (simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_set_disconnected_nodes_get_disconnected_nodes_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma set_disconnected_nodes_get_disconnected_nodes_wf_is_l_set_disconnected_nodes_get_disconnected_nodes_wf [instances]:
"l_set_disconnected_nodes_get_disconnected_nodes_wf type_wf known_ptr heap_is_wellformed parent_child_rel
get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs"
apply(auto simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf_def
l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms_def instances)[1]
using remove_from_disconnected_nodes_removes apply fast
done
subsection ‹get\_root\_node›
locale l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_heap_is_wellformed
type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
+ l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs
+ l_get_parent_wf
type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes
get_child_nodes_locs get_parent get_parent_locs
+ l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs
get_ancestors get_ancestors_locs get_root_node get_root_node_locs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
begin
lemma get_ancestors_reads:
assumes "heap_is_wellformed h"
shows "reads get_ancestors_locs (get_ancestors node_ptr) h h'"
proof (insert assms(1), induct rule: heap_wellformed_induct_rev)
case (step child)
then show ?case
using [[simproc del: Product_Type.unit_eq]] get_parent_reads[unfolded reads_def]
apply(simp (no_asm) add: get_ancestors_def)
by(auto simp add: get_ancestors_locs_def reads_subset[OF return_reads] get_parent_reads_pointers
intro!: reads_bind_pure reads_subset[OF check_in_heap_reads]
reads_subset[OF get_parent_reads] reads_subset[OF get_child_nodes_reads]
split: option.splits)
qed
lemma get_ancestors_ok:
assumes "heap_is_wellformed h"
and "ptr |∈| object_ptr_kinds h"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "h ⊢ ok (get_ancestors ptr)"
proof (insert assms(1) assms(2), induct rule: heap_wellformed_induct_rev)
case (step child)
then show ?case
using assms(3) assms(4)
apply(simp (no_asm) add: get_ancestors_def)
apply(simp add: assms(1) get_parent_parent_in_heap)
by(auto intro!: bind_is_OK_pure_I bind_pure_I get_parent_ok split: option.splits)
qed
lemma get_root_node_ptr_in_heap:
assumes "h ⊢ ok (get_root_node ptr)"
shows "ptr |∈| object_ptr_kinds h"
using assms
unfolding get_root_node_def
using get_ancestors_ptr_in_heap
by auto
lemma get_root_node_ok:
assumes "heap_is_wellformed h" "known_ptrs h" "type_wf h"
and "ptr |∈| object_ptr_kinds h"
shows "h ⊢ ok (get_root_node ptr)"
unfolding get_root_node_def
using assms get_ancestors_ok
by auto
lemma get_ancestors_parent:
assumes "heap_is_wellformed h"
and "h ⊢ get_parent child →⇩r Some parent"
shows "h ⊢ get_ancestors (cast child) →⇩r (cast child) # parent # ancestors
⟷ h ⊢ get_ancestors parent →⇩r parent # ancestors"
proof
assume a1: "h ⊢ get_ancestors (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r
cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child # parent # ancestors"
then have "h ⊢ Heap_Error_Monad.bind (check_in_heap (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child))
(λ_. Heap_Error_Monad.bind (get_parent child)
(λx. Heap_Error_Monad.bind (case x of None ⇒ return [] | Some x ⇒ get_ancestors x)
(λancestors. return (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child # ancestors))))
→⇩r cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child # parent # ancestors"
by(simp add: get_ancestors_def)
then show "h ⊢ get_ancestors parent →⇩r parent # ancestors"
using assms(2) apply(auto elim!: bind_returns_result_E2 split: option.splits)[1]
using returns_result_eq by fastforce
next
assume "h ⊢ get_ancestors parent →⇩r parent # ancestors"
then show "h ⊢ get_ancestors (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child # parent # ancestors"
using assms(2)
apply(simp (no_asm) add: get_ancestors_def)
apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1]
by (metis (full_types) assms(2) check_in_heap_ptr_in_heap is_OK_returns_result_I
local.get_parent_ptr_in_heap node_ptr_kinds_commutes old.unit.exhaust
select_result_I)
qed
lemma get_ancestors_never_empty:
assumes "heap_is_wellformed h"
and "h ⊢ get_ancestors child →⇩r ancestors"
shows "ancestors ≠ []"
proof(insert assms(2), induct arbitrary: ancestors rule: heap_wellformed_induct_rev[OF assms(1)])
case (1 child)
then show ?case
proof (induct "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child")
case None
then show ?case
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits)
next
case (Some child_node)
then obtain parent_opt where parent_opt: "h ⊢ get_parent child_node →⇩r parent_opt"
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits)
with Some show ?case
proof(induct parent_opt)
case None
then show ?case
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits)
next
case (Some option)
then show ?case
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits)
qed
qed
qed
lemma get_ancestors_subset:
assumes "heap_is_wellformed h"
and "h ⊢ get_ancestors ptr →⇩r ancestors"
and "ancestor ∈ set ancestors"
and "h ⊢ get_ancestors ancestor →⇩r ancestor_ancestors"
and type_wf: "type_wf h"
and known_ptrs: "known_ptrs h"
shows "set ancestor_ancestors ⊆ set ancestors"
proof (insert assms(1) assms(2) assms(3), induct ptr arbitrary: ancestors
rule: heap_wellformed_induct_rev)
case (step child)
have "child |∈| object_ptr_kinds h"
using get_ancestors_ptr_in_heap step(2) by auto
show ?case
proof (induct "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child")
case None
then have "ancestors = [child]"
using step(2) step(3)
by(auto simp add: get_ancestors_def elim!: bind_returns_result_E2)
show ?case
using step(2) step(3)
apply(auto simp add: ‹ancestors = [child]›)[1]
using assms(4) returns_result_eq by fastforce
next
case (Some child_node)
note s1 = Some
obtain parent_opt where parent_opt: "h ⊢ get_parent child_node →⇩r parent_opt"
using ‹child |∈| object_ptr_kinds h› assms(1) Some[symmetric]
get_parent_ok[OF type_wf known_ptrs]
by (metis (no_types, lifting) is_OK_returns_result_E known_ptrs get_parent_ok
l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms node_ptr_casts_commute node_ptr_kinds_commutes)
then show ?case
proof (induct parent_opt)
case None
then have "ancestors = [child]"
using step(2) step(3) s1
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
show ?case
using step(2) step(3)
apply(auto simp add: ‹ancestors = [child]›)[1]
using assms(4) returns_result_eq by fastforce
next
case (Some parent)
have "h ⊢ Heap_Error_Monad.bind (check_in_heap child)
(λ_. Heap_Error_Monad.bind
(case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child of None ⇒ return []
| Some node_ptr ⇒ Heap_Error_Monad.bind (get_parent node_ptr)
(λparent_ptr_opt. case parent_ptr_opt of None ⇒ return []
| Some x ⇒ get_ancestors x))
(λancestors. return (child # ancestors)))
→⇩r ancestors"
using step(2)
by(simp add: get_ancestors_def)
moreover obtain tl_ancestors where tl_ancestors: "ancestors = child # tl_ancestors"
using calculation
by(auto elim!: bind_returns_result_E2 split: option.splits)
ultimately have "h ⊢ get_ancestors parent →⇩r tl_ancestors"
using s1 Some
by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
show ?case
using step(1)[OF s1[symmetric, simplified] Some ‹h ⊢ get_ancestors parent →⇩r tl_ancestors›]
step(3)
apply(auto simp add: tl_ancestors)[1]
by (metis assms(4) insert_iff list.simps(15) local.step(2) returns_result_eq tl_ancestors)
qed
qed
qed
lemma get_ancestors_also_parent:
assumes "heap_is_wellformed h"
and "h ⊢ get_ancestors some_ptr →⇩r ancestors"
and "cast child ∈ set ancestors"
and "h ⊢ get_parent child →⇩r Some parent"
and type_wf: "type_wf h"
and known_ptrs: "known_ptrs h"
shows "parent ∈ set ancestors"
proof -
obtain child_ancestors where child_ancestors: "h ⊢ get_ancestors (cast child) →⇩r child_ancestors"
by (meson assms(1) assms(4) get_ancestors_ok is_OK_returns_result_I known_ptrs
local.get_parent_ptr_in_heap node_ptr_kinds_commutes returns_result_select_result
type_wf)
then have "parent ∈ set child_ancestors"
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits dest!: returns_result_eq[OF assms(4)]
get_ancestors_ptr)
then show ?thesis
using assms child_ancestors get_ancestors_subset by blast
qed
lemma get_ancestors_obtains_children:
assumes "heap_is_wellformed h"
and "ancestor ≠ ptr"
and "ancestor ∈ set ancestors"
and "h ⊢ get_ancestors ptr →⇩r ancestors"
and type_wf: "type_wf h"
and known_ptrs: "known_ptrs h"
obtains children ancestor_child where "h ⊢ get_child_nodes ancestor →⇩r children"
and "ancestor_child ∈ set children" and "cast ancestor_child ∈ set ancestors"
proof -
assume 0: "(⋀children ancestor_child.
h ⊢ get_child_nodes ancestor →⇩r children ⟹
ancestor_child ∈ set children ⟹ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ancestor_child ∈ set ancestors
⟹ thesis)"
have "∃child. h ⊢ get_parent child →⇩r Some ancestor ∧ cast child ∈ set ancestors"
proof (insert assms(1) assms(2) assms(3) assms(4), induct ptr arbitrary: ancestors
rule: heap_wellformed_induct_rev)
case (step child)
have "child |∈| object_ptr_kinds h"
using get_ancestors_ptr_in_heap step(4) by auto
show ?case
proof (induct "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child")
case None
then have "ancestors = [child]"
using step(3) step(4)
by(auto simp add: get_ancestors_def elim!: bind_returns_result_E2)
show ?case
using step(2) step(3) step(4)
by(auto simp add: ‹ancestors = [child]›)
next
case (Some child_node)
note s1 = Some
obtain parent_opt where parent_opt: "h ⊢ get_parent child_node →⇩r parent_opt"
using ‹child |∈| object_ptr_kinds h› assms(1) Some[symmetric]
using get_parent_ok known_ptrs type_wf
by (metis (no_types, lifting) is_OK_returns_result_E node_ptr_casts_commute
node_ptr_kinds_commutes)
then show ?case
proof (induct parent_opt)
case None
then have "ancestors = [child]"
using step(2) step(3) step(4) s1
apply(simp add: get_ancestors_def)
by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
show ?case
using step(2) step(3) step(4)
by(auto simp add: ‹ancestors = [child]›)
next
case (Some parent)
have "h ⊢ Heap_Error_Monad.bind (check_in_heap child)
(λ_. Heap_Error_Monad.bind
(case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child of None ⇒ return []
| Some node_ptr ⇒ Heap_Error_Monad.bind (get_parent node_ptr)
(λparent_ptr_opt. case parent_ptr_opt of None ⇒ return []
| Some x ⇒ get_ancestors x))
(λancestors. return (child # ancestors)))
→⇩r ancestors"
using step(4)
by(simp add: get_ancestors_def)
moreover obtain tl_ancestors where tl_ancestors: "ancestors = child # tl_ancestors"
using calculation
by(auto elim!: bind_returns_result_E2 split: option.splits)
ultimately have "h ⊢ get_ancestors parent →⇩r tl_ancestors"
using s1 Some
by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
have "ancestor ∈ set tl_ancestors"
using tl_ancestors step(2) step(3) by auto
show ?case
proof (cases "ancestor ≠ parent")
case True
show ?thesis
using step(1)[OF s1[symmetric, simplified] Some True
‹ancestor ∈ set tl_ancestors› ‹h ⊢ get_ancestors parent →⇩r tl_ancestors›]
using tl_ancestors by auto
next
case False
have "child ∈ set ancestors"
using step(4) get_ancestors_ptr by simp
then show ?thesis
using Some False s1[symmetric] by(auto)
qed
qed
qed
qed
then obtain child where child: "h ⊢ get_parent child →⇩r Some ancestor"
and in_ancestors: "cast child ∈ set ancestors"
by auto
then obtain children where
children: "h ⊢ get_child_nodes ancestor →⇩r children" and
child_in_children: "child ∈ set children"
using get_parent_child_dual by blast
show thesis
using 0[OF children child_in_children] child assms(3) in_ancestors by blast
qed
lemma get_ancestors_parent_child_rel:
assumes "heap_is_wellformed h"
and "h ⊢ get_ancestors child →⇩r ancestors"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "(ptr, child) ∈ (parent_child_rel h)⇧* ⟷ ptr ∈ set ancestors"
proof (safe)
assume 3: "(ptr, child) ∈ (parent_child_rel h)⇧*"
show "ptr ∈ set ancestors"
proof (insert 3, induct ptr rule: heap_wellformed_induct[OF assms(1)])
case (1 ptr)
then show ?case
proof (cases "ptr = child")
case True
then show ?thesis
by (metis (no_types, lifting) assms(2) bind_returns_result_E get_ancestors_def
in_set_member member_rec(1) return_returns_result)
next
case False
obtain ptr_child where
ptr_child: "(ptr, ptr_child) ∈ (parent_child_rel h) ∧ (ptr_child, child) ∈ (parent_child_rel h)⇧*"
using converse_rtranclE[OF 1(2)] ‹ptr ≠ child›
by metis
then obtain ptr_child_node
where ptr_child_ptr_child_node: "ptr_child = cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr_child_node"
using ptr_child node_ptr_casts_commute3 parent_child_rel_node_ptr
by (metis )
then obtain children where
children: "h ⊢ get_child_nodes ptr →⇩r children" and
ptr_child_node: "ptr_child_node ∈ set children"
proof -
assume a1: "⋀children. ⟦h ⊢ get_child_nodes ptr →⇩r children; ptr_child_node ∈ set children⟧
⟹ thesis"
have "ptr |∈| object_ptr_kinds h"
using local.parent_child_rel_parent_in_heap ptr_child by blast
moreover have "ptr_child_node ∈ set |h ⊢ get_child_nodes ptr|⇩r"
by (metis calculation known_ptrs local.get_child_nodes_ok local.known_ptrs_known_ptr
local.parent_child_rel_child ptr_child ptr_child_ptr_child_node
returns_result_select_result type_wf)
ultimately show ?thesis
using a1 get_child_nodes_ok type_wf known_ptrs
by (meson local.known_ptrs_known_ptr returns_result_select_result)
qed
moreover have "(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr_child_node, child) ∈ (parent_child_rel h)⇧*"
using ptr_child ptr_child_ptr_child_node by auto
ultimately have "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr_child_node ∈ set ancestors"
using 1 by auto
moreover have "h ⊢ get_parent ptr_child_node →⇩r Some ptr"
using assms(1) children ptr_child_node child_parent_dual
using known_ptrs type_wf by blast
ultimately show ?thesis
using get_ancestors_also_parent assms type_wf by blast
qed
qed
next
assume 3: "ptr ∈ set ancestors"
show "(ptr, child) ∈ (parent_child_rel h)⇧*"
proof (insert 3, induct ptr rule: heap_wellformed_induct[OF assms(1)])
case (1 ptr)
then show ?case
proof (cases "ptr = child")
case True
then show ?thesis
by simp
next
case False
then obtain children ptr_child_node where
children: "h ⊢ get_child_nodes ptr →⇩r children" and
ptr_child_node: "ptr_child_node ∈ set children" and
ptr_child_node_in_ancestors: "cast ptr_child_node ∈ set ancestors"
using 1(2) assms(2) get_ancestors_obtains_children assms(1)
using known_ptrs type_wf by blast
then have "(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr_child_node, child) ∈ (parent_child_rel h)⇧*"
using 1(1) by blast
moreover have "(ptr, cast ptr_child_node) ∈ parent_child_rel h"
using children ptr_child_node assms(1) parent_child_rel_child_nodes2
using child_parent_dual known_ptrs parent_child_rel_parent type_wf
by blast
ultimately show ?thesis
by auto
qed
qed
qed
lemma get_root_node_parent_child_rel:
assumes "heap_is_wellformed h"
and "h ⊢ get_root_node child →⇩r root"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "(root, child) ∈ (parent_child_rel h)⇧*"
using assms get_ancestors_parent_child_rel
apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2)[1]
using get_ancestors_never_empty last_in_set by blast
lemma get_ancestors_eq:
assumes "heap_is_wellformed h"
and "heap_is_wellformed h'"
and "⋀object_ptr w. object_ptr ≠ ptr ⟹ w ∈ get_child_nodes_locs object_ptr ⟹ w h h'"
and pointers_preserved: "⋀object_ptr. preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t object_ptr RObject.nothing) h h'"
and known_ptrs: "known_ptrs h"
and known_ptrs': "known_ptrs h'"
and "h ⊢ get_ancestors ptr →⇩r ancestors"
and type_wf: "type_wf h"
and type_wf': "type_wf h'"
shows "h' ⊢ get_ancestors ptr →⇩r ancestors"
proof -
have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
using pointers_preserved object_ptr_kinds_preserved_small by blast
then have object_ptr_kinds_M_eq:
"⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_eq: "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by(simp)
have "h' ⊢ ok (get_ancestors ptr)"
using get_ancestors_ok get_ancestors_ptr_in_heap object_ptr_kinds_eq3 assms(1) known_ptrs
known_ptrs' assms(2) assms(7) type_wf'
by blast
then obtain ancestors' where ancestors': "h' ⊢ get_ancestors ptr →⇩r ancestors'"
by auto
obtain root where root: "h ⊢ get_root_node ptr →⇩r root"
proof -
assume 0: "(⋀root. h ⊢ get_root_node ptr →⇩r root ⟹ thesis)"
show thesis
apply(rule 0)
using assms(7)
by(auto simp add: get_root_node_def elim!: bind_returns_result_E2 split: option.splits)
qed
have children_eq:
"⋀p children. p ≠ ptr ⟹ h ⊢ get_child_nodes p →⇩r children = h' ⊢ get_child_nodes p →⇩r children"
using get_child_nodes_reads assms(3)
apply(simp add: reads_def reflp_def transp_def preserved_def)
by blast
have "acyclic (parent_child_rel h)"
using assms(1) local.parent_child_rel_acyclic by auto
have "acyclic (parent_child_rel h')"
using assms(2) local.parent_child_rel_acyclic by blast
have 2: "⋀c parent_opt. cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c ∈ set ancestors ∩ set ancestors'
⟹ h ⊢ get_parent c →⇩r parent_opt = h' ⊢ get_parent c →⇩r parent_opt"
proof -
fix c parent_opt
assume 1: " cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c ∈ set ancestors ∩ set ancestors'"
obtain ptrs where ptrs: "h ⊢ object_ptr_kinds_M →⇩r ptrs"
by simp
let ?P = "(λptr. Heap_Error_Monad.bind (get_child_nodes ptr) (λchildren. return (c ∈ set children)))"
have children_eq_True: "⋀p. p ∈ set ptrs ⟹ h ⊢ ?P p →⇩r True ⟷ h' ⊢ ?P p →⇩r True"
proof -
fix p
assume "p ∈ set ptrs"
then show "h ⊢ ?P p →⇩r True ⟷ h' ⊢ ?P p →⇩r True"
proof (cases "p = ptr")
case True
have "(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c, ptr) ∈ (parent_child_rel h)⇧*"
using get_ancestors_parent_child_rel 1 assms by blast
then have "(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h)"
proof (cases "cast c = ptr")
case True
then show ?thesis
using ‹acyclic (parent_child_rel h)› by(auto simp add: acyclic_def)
next
case False
then have "(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h)⇧*"
using ‹acyclic (parent_child_rel h)› False rtrancl_eq_or_trancl rtrancl_trancl_trancl
‹(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c, ptr) ∈ (parent_child_rel h)⇧*›
by (metis acyclic_def)
then show ?thesis
using r_into_rtrancl by auto
qed
obtain children where children: "h ⊢ get_child_nodes ptr →⇩r children"
using type_wf
by (metis ‹h' ⊢ ok get_ancestors ptr› assms(1) get_ancestors_ptr_in_heap get_child_nodes_ok
heap_is_wellformed_def is_OK_returns_result_E known_ptrs local.known_ptrs_known_ptr
object_ptr_kinds_eq3)
then have "c ∉ set children"
using ‹(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h)› assms(1)
using parent_child_rel_child_nodes2
using child_parent_dual known_ptrs parent_child_rel_parent
type_wf by blast
with children have "h ⊢ ?P p →⇩r False"
by(auto simp add: True)
moreover have "(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c, ptr) ∈ (parent_child_rel h')⇧*"
using get_ancestors_parent_child_rel assms(2) ancestors' 1 known_ptrs' type_wf
type_wf' by blast
then have "(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h')"
proof (cases "cast c = ptr")
case True
then show ?thesis
using ‹acyclic (parent_child_rel h')› by(auto simp add: acyclic_def)
next
case False
then have "(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h')⇧*"
using ‹acyclic (parent_child_rel h')› False rtrancl_eq_or_trancl rtrancl_trancl_trancl
‹(cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c, ptr) ∈ (parent_child_rel h')⇧*›
by (metis acyclic_def)
then show ?thesis
using r_into_rtrancl by auto
qed
then have "(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h')"
using r_into_rtrancl by auto
obtain children' where children': "h' ⊢ get_child_nodes ptr →⇩r children'"
using type_wf type_wf'
by (meson ‹h' ⊢ ok (get_ancestors ptr)› assms(2) get_ancestors_ptr_in_heap
get_child_nodes_ok is_OK_returns_result_E known_ptrs'
local.known_ptrs_known_ptr)
then have "c ∉ set children'"
using ‹(ptr, cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) ∉ (parent_child_rel h')› assms(2) type_wf type_wf'
using parent_child_rel_child_nodes2 child_parent_dual known_ptrs' parent_child_rel_parent
by auto
with children' have "h' ⊢ ?P p →⇩r False"
by(auto simp add: True)
ultimately show ?thesis
by (metis returns_result_eq)
next
case False
then show ?thesis
using children_eq ptrs
by (metis (no_types, lifting) bind_pure_returns_result_I bind_returns_result_E
get_child_nodes_pure return_returns_result)
qed
qed
have "⋀pa. pa ∈ set ptrs ⟹ h ⊢ ok (get_child_nodes pa
⤜ (λchildren. return (c ∈ set children))) = h' ⊢ ok ( get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)))"
using assms(1) assms(2) object_ptr_kinds_eq ptrs type_wf type_wf'
by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M bind_is_OK_pure_I
get_child_nodes_ok get_child_nodes_pure known_ptrs'
local.known_ptrs_known_ptr return_ok select_result_I2)
have children_eq_False:
"⋀pa. pa ∈ set ptrs ⟹ h ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r False = h' ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r False"
proof
fix pa
assume "pa ∈ set ptrs"
and "h ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
have "h ⊢ ok (get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)))
⟹ h' ⊢ ok ( get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)))"
using ‹pa ∈ set ptrs› ‹⋀pa. pa ∈ set ptrs ⟹ h ⊢ ok (get_child_nodes pa
⤜ (λchildren. return (c ∈ set children))) = h' ⊢ ok ( get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)))›
by auto
moreover have "h ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False
⟹ h' ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
by (metis (mono_tags, lifting) ‹⋀pa. pa ∈ set ptrs
⟹ h ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r True = h' ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r True› ‹pa ∈ set ptrs›
calculation is_OK_returns_result_I returns_result_eq returns_result_select_result)
ultimately show "h' ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
using ‹h ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False›
by auto
next
fix pa
assume "pa ∈ set ptrs"
and "h' ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
have "h' ⊢ ok (get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)))
⟹ h ⊢ ok ( get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)))"
using ‹pa ∈ set ptrs› ‹⋀pa. pa ∈ set ptrs
⟹ h ⊢ ok (get_child_nodes pa
⤜ (λchildren. return (c ∈ set children))) = h' ⊢ ok ( get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)))›
by auto
moreover have "h' ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False
⟹ h ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
by (metis (mono_tags, lifting)
‹⋀pa. pa ∈ set ptrs ⟹ h ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r True = h' ⊢ get_child_nodes pa
⤜ (λchildren. return (c ∈ set children)) →⇩r True› ‹pa ∈ set ptrs›
calculation is_OK_returns_result_I returns_result_eq returns_result_select_result)
ultimately show "h ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False"
using ‹h' ⊢ get_child_nodes pa ⤜ (λchildren. return (c ∈ set children)) →⇩r False› by blast
qed
have filter_eq: "⋀xs. h ⊢ filter_M ?P ptrs →⇩r xs = h' ⊢ filter_M ?P ptrs →⇩r xs"
proof (rule filter_M_eq)
show
"⋀xs x. pure (Heap_Error_Monad.bind (get_child_nodes x) (λchildren. return (c ∈ set children))) h"
by(auto intro!: bind_pure_I)
next
show
"⋀xs x. pure (Heap_Error_Monad.bind (get_child_nodes x) (λchildren. return (c ∈ set children))) h'"
by(auto intro!: bind_pure_I)
next
fix xs b x
assume 0: "x ∈ set ptrs"
then show "h ⊢ Heap_Error_Monad.bind (get_child_nodes x) (λchildren. return (c ∈ set children)) →⇩r b
= h' ⊢ Heap_Error_Monad.bind (get_child_nodes x) (λchildren. return (c ∈ set children)) →⇩r b"
apply(induct b)
using children_eq_True apply blast
using children_eq_False apply blast
done
qed
show "h ⊢ get_parent c →⇩r parent_opt = h' ⊢ get_parent c →⇩r parent_opt"
apply(simp add: get_parent_def)
apply(rule bind_cong_2)
apply(simp)
apply(simp)
apply(simp add: check_in_heap_def node_ptr_kinds_def object_ptr_kinds_eq3)
apply(rule bind_cong_2)
apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1]
apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1]
apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1]
apply(rule bind_cong_2)
apply(auto intro!: filter_M_pure_I bind_pure_I)[1]
apply(auto intro!: filter_M_pure_I bind_pure_I)[1]
apply(auto simp add: filter_eq )[1]
using filter_eq ptrs apply auto[1]
using filter_eq ptrs by auto
qed
have "ancestors = ancestors'"
proof(insert assms(1) assms(7) ancestors' 2, induct ptr arbitrary: ancestors ancestors'
rule: heap_wellformed_induct_rev)
case (step child)
show ?case
using step(2) step(3) step(4)
apply(simp add: get_ancestors_def)
apply(auto intro!: elim!: bind_returns_result_E2 split: option.splits)[1]
using returns_result_eq apply fastforce
apply (meson option.simps(3) returns_result_eq)
by (metis IntD1 IntD2 option.inject returns_result_eq step.hyps)
qed
then show ?thesis
using assms(5) ancestors'
by simp
qed
lemma get_ancestors_remains_not_in_ancestors:
assumes "heap_is_wellformed h"
and "heap_is_wellformed h'"
and "h ⊢ get_ancestors ptr →⇩r ancestors"
and "h' ⊢ get_ancestors ptr →⇩r ancestors'"
and "⋀p children children'. h ⊢ get_child_nodes p →⇩r children
⟹ h' ⊢ get_child_nodes p →⇩r children' ⟹ set children' ⊆ set children"
and "node ∉ set ancestors"
and object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
and type_wf': "type_wf h'"
shows "node ∉ set ancestors'"
proof -
have object_ptr_kinds_M_eq:
"⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
using object_ptr_kinds_eq3
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_eq: "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by(simp)
show ?thesis
proof (insert assms(1) assms(3) assms(4) assms(6), induct ptr arbitrary: ancestors ancestors'
rule: heap_wellformed_induct_rev)
case (step child)
have 1: "⋀p parent. h' ⊢ get_parent p →⇩r Some parent ⟹ h ⊢ get_parent p →⇩r Some parent"
proof -
fix p parent
assume "h' ⊢ get_parent p →⇩r Some parent"
then obtain children' where
children': "h' ⊢ get_child_nodes parent →⇩r children'" and
p_in_children': "p ∈ set children'"
using get_parent_child_dual by blast
obtain children where children: "h ⊢ get_child_nodes parent →⇩r children"
using get_child_nodes_ok assms(1) get_child_nodes_ptr_in_heap object_ptr_kinds_eq children'
known_ptrs
using type_wf type_wf'
by (metis ‹h' ⊢ get_parent p →⇩r Some parent› get_parent_parent_in_heap is_OK_returns_result_E
local.known_ptrs_known_ptr object_ptr_kinds_eq3)
have "p ∈ set children"
using assms(5) children children' p_in_children'
by blast
then show "h ⊢ get_parent p →⇩r Some parent"
using child_parent_dual assms(1) children known_ptrs type_wf by blast
qed
have "node ≠ child"
using assms(1) get_ancestors_parent_child_rel step.prems(1) step.prems(3) known_ptrs
using type_wf type_wf'
by blast
then show ?case
using step(2) step(3)
apply(simp add: get_ancestors_def)
using step(4)
apply(auto elim!: bind_returns_result_E2 split: option.splits)[1]
using 1
apply (meson option.distinct(1) returns_result_eq)
by (metis "1" option.inject returns_result_eq step.hyps)
qed
qed
lemma get_ancestors_ptrs_in_heap:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_ancestors ptr →⇩r ancestors"
assumes "ptr' ∈ set ancestors"
shows "ptr' |∈| object_ptr_kinds h"
proof (insert assms(4) assms(5), induct ancestors arbitrary: ptr)
case Nil
then show ?case
by(auto)
next
case (Cons a ancestors)
then obtain x where x: "h ⊢ get_ancestors x →⇩r a # ancestors"
by(auto simp add: get_ancestors_def[of a] elim!: bind_returns_result_E2 split: option.splits)
then have "x = a"
by(auto simp add: get_ancestors_def[of x] elim!: bind_returns_result_E2 split: option.splits)
then show ?case
using Cons.hyps Cons.prems(2) get_ancestors_ptr_in_heap x
by (metis assms(1) assms(2) assms(3) get_ancestors_obtains_children get_child_nodes_ptr_in_heap
is_OK_returns_result_I)
qed
lemma get_ancestors_prefix:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_ancestors ptr →⇩r ancestors"
assumes "ptr' ∈ set ancestors"
assumes "h ⊢ get_ancestors ptr' →⇩r ancestors'"
shows "∃pre. ancestors = pre @ ancestors'"
proof (insert assms(1) assms(5) assms(6), induct ptr' arbitrary: ancestors'
rule: heap_wellformed_induct)
case (step parent)
then show ?case
proof (cases "parent ≠ ptr" )
case True
then obtain children ancestor_child where "h ⊢ get_child_nodes parent →⇩r children"
and "ancestor_child ∈ set children" and "cast ancestor_child ∈ set ancestors"
using assms(1) assms(2) assms(3) assms(4) get_ancestors_obtains_children step.prems(1) by blast
then have "h ⊢ get_parent ancestor_child →⇩r Some parent"
using assms(1) assms(2) assms(3) child_parent_dual by blast
then have "h ⊢ get_ancestors (cast ancestor_child) →⇩r cast ancestor_child # ancestors'"
apply(simp add: get_ancestors_def)
using ‹h ⊢ get_ancestors parent →⇩r ancestors'› get_parent_ptr_in_heap
by(auto simp add: check_in_heap_def is_OK_returns_result_I intro!: bind_pure_returns_result_I)
then show ?thesis
using step(1) ‹h ⊢ get_child_nodes parent →⇩r children› ‹ancestor_child ∈ set children›
‹cast ancestor_child ∈ set ancestors›
‹h ⊢ get_ancestors (cast ancestor_child) →⇩r cast ancestor_child # ancestors'›
by fastforce
next
case False
then show ?thesis
by (metis append_Nil assms(4) returns_result_eq step.prems(2))
qed
qed
lemma get_ancestors_same_root_node:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_ancestors ptr →⇩r ancestors"
assumes "ptr' ∈ set ancestors"
assumes "ptr'' ∈ set ancestors"
shows "h ⊢ get_root_node ptr' →⇩r root_ptr ⟷ h ⊢ get_root_node ptr'' →⇩r root_ptr"
proof -
have "ptr' |∈| object_ptr_kinds h"
by (metis assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_obtains_children
get_ancestors_ptr_in_heap get_child_nodes_ptr_in_heap is_OK_returns_result_I)
then obtain ancestors' where ancestors': "h ⊢ get_ancestors ptr' →⇩r ancestors'"
by (meson assms(1) assms(2) assms(3) get_ancestors_ok is_OK_returns_result_E)
then have "∃pre. ancestors = pre @ ancestors'"
using get_ancestors_prefix assms by blast
moreover have "ptr'' |∈| object_ptr_kinds h"
by (metis assms(1) assms(2) assms(3) assms(4) assms(6) get_ancestors_obtains_children
get_ancestors_ptr_in_heap get_child_nodes_ptr_in_heap is_OK_returns_result_I)
then obtain ancestors'' where ancestors'': "h ⊢ get_ancestors ptr'' →⇩r ancestors''"
by (meson assms(1) assms(2) assms(3) get_ancestors_ok is_OK_returns_result_E)
then have "∃pre. ancestors = pre @ ancestors''"
using get_ancestors_prefix assms by blast
ultimately show ?thesis
using ancestors' ancestors''
apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I)[1]
apply (metis (no_types, lifting) assms(1) get_ancestors_never_empty last_appendR
returns_result_eq)
by (metis assms(1) get_ancestors_never_empty last_appendR returns_result_eq)
qed
lemma get_root_node_parent_same:
assumes "h ⊢ get_parent child →⇩r Some ptr"
shows "h ⊢ get_root_node (cast child) →⇩r root ⟷ h ⊢ get_root_node ptr →⇩r root"
proof
assume 1: " h ⊢ get_root_node (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r root"
show "h ⊢ get_root_node ptr →⇩r root"
using 1[unfolded get_root_node_def] assms
apply(simp add: get_ancestors_def)
apply(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I split: option.splits)[1]
using returns_result_eq apply fastforce
using get_ancestors_ptr by fastforce
next
assume 1: " h ⊢ get_root_node ptr →⇩r root"
show "h ⊢ get_root_node (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r root"
apply(simp add: get_root_node_def)
using assms 1
apply(simp add: get_ancestors_def)
apply(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I split: option.splits)[1]
apply (simp add: check_in_heap_def is_OK_returns_result_I)
using get_ancestors_ptr get_parent_ptr_in_heap
apply (simp add: is_OK_returns_result_I)
by (meson list.distinct(1) list.set_cases local.get_ancestors_ptr)
qed
lemma get_root_node_same_no_parent:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r cast child"
shows "h ⊢ get_parent child →⇩r None"
proof (insert assms(1) assms(4), induct ptr rule: heap_wellformed_induct_rev)
case (step c)
then show ?case
proof (cases "cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r c")
case None
then have "c = cast child"
using step(2)
by(auto simp add: get_root_node_def get_ancestors_def[of c] elim!: bind_returns_result_E2)
then show ?thesis
using None by auto
next
case (Some child_node)
note s = this
then obtain parent_opt where parent_opt: "h ⊢ get_parent child_node →⇩r parent_opt"
by (metis (no_types, lifting) assms(2) assms(3) get_root_node_ptr_in_heap
is_OK_returns_result_I local.get_parent_ok node_ptr_casts_commute
node_ptr_kinds_commutes returns_result_select_result step.prems)
then show ?thesis
proof(induct parent_opt)
case None
then show ?case
using Some get_root_node_no_parent returns_result_eq step.prems by fastforce
next
case (Some parent)
then show ?case
using step s
apply(auto simp add: get_root_node_def get_ancestors_def[of c]
elim!: bind_returns_result_E2 split: option.splits list.splits)[1]
using get_root_node_parent_same step.hyps step.prems by auto
qed
qed
qed
lemma get_root_node_not_node_same:
assumes "ptr |∈| object_ptr_kinds h"
assumes "¬is_node_ptr_kind ptr"
shows "h ⊢ get_root_node ptr →⇩r ptr"
using assms
apply(simp add: get_root_node_def get_ancestors_def)
by(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I split: option.splits)
lemma get_root_node_root_in_heap:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r root"
shows "root |∈| object_ptr_kinds h"
using assms
apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2)[1]
by (simp add: get_ancestors_never_empty get_ancestors_ptrs_in_heap)
lemma get_root_node_same_no_parent_parent_child_rel:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr' →⇩r ptr'"
shows "¬(∃p. (p, ptr') ∈ (parent_child_rel h))"
by (metis (no_types, lifting) assms(1) assms(2) assms(3) assms(4) get_root_node_same_no_parent
l_heap_is_wellformed.parent_child_rel_child local.child_parent_dual local.get_child_nodes_ok
local.known_ptrs_known_ptr local.l_heap_is_wellformed_axioms local.parent_child_rel_node_ptr
local.parent_child_rel_parent_in_heap node_ptr_casts_commute3 option.simps(3) returns_result_eq
returns_result_select_result)
end
locale l_get_ancestors_wf = l_heap_is_wellformed_defs + l_known_ptrs + l_type_wf + l_get_ancestors_defs
+ l_get_child_nodes_defs + l_get_parent_defs +
assumes get_ancestors_never_empty:
"heap_is_wellformed h ⟹ h ⊢ get_ancestors child →⇩r ancestors ⟹ ancestors ≠ []"
assumes get_ancestors_ok:
"heap_is_wellformed h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptrs h ⟹ type_wf h
⟹ h ⊢ ok (get_ancestors ptr)"
assumes get_ancestors_reads:
"heap_is_wellformed h ⟹ reads get_ancestors_locs (get_ancestors node_ptr) h h'"
assumes get_ancestors_ptrs_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_ancestors ptr →⇩r ancestors ⟹ ptr' ∈ set ancestors
⟹ ptr' |∈| object_ptr_kinds h"
assumes get_ancestors_remains_not_in_ancestors:
"heap_is_wellformed h ⟹ heap_is_wellformed h' ⟹ h ⊢ get_ancestors ptr →⇩r ancestors
⟹ h' ⊢ get_ancestors ptr →⇩r ancestors'
⟹ (⋀p children children'. h ⊢ get_child_nodes p →⇩r children
⟹ h' ⊢ get_child_nodes p →⇩r children'
⟹ set children' ⊆ set children)
⟹ node ∉ set ancestors
⟹ object_ptr_kinds h = object_ptr_kinds h' ⟹ known_ptrs h
⟹ type_wf h ⟹ type_wf h' ⟹ node ∉ set ancestors'"
assumes get_ancestors_also_parent:
"heap_is_wellformed h ⟹ h ⊢ get_ancestors some_ptr →⇩r ancestors
⟹ cast child_node ∈ set ancestors
⟹ h ⊢ get_parent child_node →⇩r Some parent ⟹ type_wf h
⟹ known_ptrs h ⟹ parent ∈ set ancestors"
assumes get_ancestors_obtains_children:
"heap_is_wellformed h ⟹ ancestor ≠ ptr ⟹ ancestor ∈ set ancestors
⟹ h ⊢ get_ancestors ptr →⇩r ancestors ⟹ type_wf h ⟹ known_ptrs h
⟹ (⋀children ancestor_child . h ⊢ get_child_nodes ancestor →⇩r children
⟹ ancestor_child ∈ set children
⟹ cast ancestor_child ∈ set ancestors
⟹ thesis)
⟹ thesis"
assumes get_ancestors_parent_child_rel:
"heap_is_wellformed h ⟹ h ⊢ get_ancestors child →⇩r ancestors ⟹ known_ptrs h ⟹ type_wf h
⟹ (ptr, child) ∈ (parent_child_rel h)⇧* ⟷ ptr ∈ set ancestors"
locale l_get_root_node_wf = l_heap_is_wellformed_defs + l_get_root_node_defs + l_type_wf
+ l_known_ptrs + l_get_ancestors_defs + l_get_parent_defs +
assumes get_root_node_ok:
"heap_is_wellformed h ⟹ known_ptrs h ⟹ type_wf h ⟹ ptr |∈| object_ptr_kinds h
⟹ h ⊢ ok (get_root_node ptr)"
assumes get_root_node_ptr_in_heap:
"h ⊢ ok (get_root_node ptr) ⟹ ptr |∈| object_ptr_kinds h"
assumes get_root_node_root_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_root_node ptr →⇩r root ⟹ root |∈| object_ptr_kinds h"
assumes get_ancestors_same_root_node:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_ancestors ptr →⇩r ancestors ⟹ ptr' ∈ set ancestors
⟹ ptr'' ∈ set ancestors
⟹ h ⊢ get_root_node ptr' →⇩r root_ptr ⟷ h ⊢ get_root_node ptr'' →⇩r root_ptr"
assumes get_root_node_same_no_parent:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_root_node ptr →⇩r cast child ⟹ h ⊢ get_parent child →⇩r None"
assumes get_root_node_parent_same:
"h ⊢ get_parent child →⇩r Some ptr
⟹ h ⊢ get_root_node (cast child) →⇩r root ⟷ h ⊢ get_root_node ptr →⇩r root"
interpretation i_get_root_node_wf?:
l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel
get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs
using instances
by(simp add: l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
lemma get_ancestors_wf_is_l_get_ancestors_wf [instances]:
"l_get_ancestors_wf heap_is_wellformed parent_child_rel known_ptr known_ptrs type_wf get_ancestors
get_ancestors_locs get_child_nodes get_parent"
using known_ptrs_is_l_known_ptrs
apply(auto simp add: l_get_ancestors_wf_def l_get_ancestors_wf_axioms_def)[1]
using get_ancestors_never_empty apply blast
using get_ancestors_ok apply blast
using get_ancestors_reads apply blast
using get_ancestors_ptrs_in_heap apply blast
using get_ancestors_remains_not_in_ancestors apply blast
using get_ancestors_also_parent apply blast
using get_ancestors_obtains_children apply blast
using get_ancestors_parent_child_rel apply blast
using get_ancestors_parent_child_rel apply blast
done
lemma get_root_node_wf_is_l_get_root_node_wf [instances]:
"l_get_root_node_wf heap_is_wellformed get_root_node type_wf known_ptr known_ptrs
get_ancestors get_parent"
using known_ptrs_is_l_known_ptrs
apply(auto simp add: l_get_root_node_wf_def l_get_root_node_wf_axioms_def)[1]
using get_root_node_ok apply blast
using get_root_node_ptr_in_heap apply blast
using get_root_node_root_in_heap apply blast
using get_ancestors_same_root_node apply(blast, blast)
using get_root_node_same_no_parent apply blast
using get_root_node_parent_same apply (blast, blast)
done
subsection ‹to\_tree\_order›
locale l_to_tree_order_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent +
l_get_parent_wf +
l_heap_is_wellformed
begin
lemma to_tree_order_ptr_in_heap:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ ok (to_tree_order ptr)"
shows "ptr |∈| object_ptr_kinds h"
proof(insert assms(1) assms(4), induct rule: heap_wellformed_induct)
case (step parent)
then show ?case
apply(auto simp add: to_tree_order_def[of parent] map_M_pure_I elim!: bind_is_OK_E3)[1]
using get_child_nodes_ptr_in_heap by blast
qed
lemma to_tree_order_either_ptr_or_in_children:
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
and "node ∈ set nodes"
and "h ⊢ get_child_nodes ptr →⇩r children"
and "node ≠ ptr"
obtains child child_to where "child ∈ set children"
and "h ⊢ to_tree_order (cast child) →⇩r child_to" and "node ∈ set child_to"
proof -
obtain treeorders where treeorders: "h ⊢ map_M to_tree_order (map cast children) →⇩r treeorders"
using assms
apply(auto simp add: to_tree_order_def elim!: bind_returns_result_E)[1]
using pure_returns_heap_eq returns_result_eq by fastforce
then have "node ∈ set (concat treeorders)"
using assms[simplified to_tree_order_def]
by(auto elim!: bind_returns_result_E4 dest: pure_returns_heap_eq)
then obtain treeorder where "treeorder ∈ set treeorders"
and node_in_treeorder: "node ∈ set treeorder"
by auto
then obtain child where "h ⊢ to_tree_order (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r treeorder"
and "child ∈ set children"
using assms[simplified to_tree_order_def] treeorders
by(auto elim!: map_M_pure_E2)
then show ?thesis
using node_in_treeorder returns_result_eq that by auto
qed
lemma to_tree_order_ptrs_in_heap:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r to"
assumes "ptr' ∈ set to"
shows "ptr' |∈| object_ptr_kinds h"
proof(insert assms(1) assms(4) assms(5), induct ptr arbitrary: to rule: heap_wellformed_induct)
case (step parent)
have "parent |∈| object_ptr_kinds h"
using assms(1) assms(2) assms(3) step.prems(1) to_tree_order_ptr_in_heap by blast
then obtain children where children: "h ⊢ get_child_nodes parent →⇩r children"
by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr)
then show ?case
proof (cases "children = []")
case True
then have "to = [parent]"
using step(2) children
apply(auto simp add: to_tree_order_def[of parent] map_M_pure_I elim!: bind_returns_result_E2)[1]
by (metis list.distinct(1) list.map_disc_iff list.set_cases map_M_pure_E2 returns_result_eq)
then show ?thesis
using ‹parent |∈| object_ptr_kinds h› step.prems(2) by auto
next
case False
note f = this
then show ?thesis
using children step to_tree_order_either_ptr_or_in_children
proof (cases "ptr' = parent")
case True
then show ?thesis
using ‹parent |∈| object_ptr_kinds h› by blast
next
case False
then show ?thesis
using children step.hyps to_tree_order_either_ptr_or_in_children
by (metis step.prems(1) step.prems(2))
qed
qed
qed
lemma to_tree_order_ok:
assumes wellformed: "heap_is_wellformed h"
and "ptr |∈| object_ptr_kinds h"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "h ⊢ ok (to_tree_order ptr)"
proof(insert assms(1) assms(2), induct rule: heap_wellformed_induct)
case (step parent)
then show ?case
using assms(3) type_wf
apply(simp add: to_tree_order_def)
apply(auto simp add: heap_is_wellformed_def intro!: map_M_ok_I bind_is_OK_pure_I map_M_pure_I)[1]
using get_child_nodes_ok known_ptrs_known_ptr apply blast
by (simp add: local.heap_is_wellformed_children_in_heap local.to_tree_order_def wellformed)
qed
lemma to_tree_order_child_subset:
assumes "heap_is_wellformed h"
and "h ⊢ to_tree_order ptr →⇩r nodes"
and "h ⊢ get_child_nodes ptr →⇩r children"
and "node ∈ set children"
and "h ⊢ to_tree_order (cast node) →⇩r nodes'"
shows "set nodes' ⊆ set nodes"
proof
fix x
assume a1: "x ∈ set nodes'"
moreover obtain treeorders
where treeorders: "h ⊢ map_M to_tree_order (map cast children) →⇩r treeorders"
using assms(2) assms(3)
apply(auto simp add: to_tree_order_def elim!: bind_returns_result_E)[1]
using pure_returns_heap_eq returns_result_eq by fastforce
then have "nodes' ∈ set treeorders"
using assms(4) assms(5)
by(auto elim!: map_M_pure_E dest: returns_result_eq)
moreover have "set (concat treeorders) ⊆ set nodes"
using treeorders assms(2) assms(3)
by(auto simp add: to_tree_order_def elim!: bind_returns_result_E4 dest: pure_returns_heap_eq)
ultimately show "x ∈ set nodes"
by auto
qed
lemma to_tree_order_ptr_in_result:
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
shows "ptr ∈ set nodes"
using assms
apply(simp add: to_tree_order_def)
by(auto elim!: bind_returns_result_E2 intro!: map_M_pure_I bind_pure_I)
lemma to_tree_order_subset:
assumes "heap_is_wellformed h"
and "h ⊢ to_tree_order ptr →⇩r nodes"
and "node ∈ set nodes"
and "h ⊢ to_tree_order node →⇩r nodes'"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "set nodes' ⊆ set nodes"
proof -
have "∀nodes. h ⊢ to_tree_order ptr →⇩r nodes ⟶ (∀node. node ∈ set nodes
⟶ (∀nodes'. h ⊢ to_tree_order node →⇩r nodes' ⟶ set nodes' ⊆ set nodes))"
proof(insert assms(1), induct ptr rule: heap_wellformed_induct)
case (step parent)
then show ?case
proof safe
fix nodes node nodes' x
assume 1: "(⋀children child.
h ⊢ get_child_nodes parent →⇩r children ⟹
child ∈ set children ⟹ ∀nodes. h ⊢ to_tree_order (cast child) →⇩r nodes
⟶ (∀node. node ∈ set nodes ⟶ (∀nodes'. h ⊢ to_tree_order node →⇩r nodes'
⟶ set nodes' ⊆ set nodes)))"
and 2: "h ⊢ to_tree_order parent →⇩r nodes"
and 3: "node ∈ set nodes"
and "h ⊢ to_tree_order node →⇩r nodes'"
and "x ∈ set nodes'"
have h1: "(⋀children child nodes node nodes'.
h ⊢ get_child_nodes parent →⇩r children ⟹
child ∈ set children ⟹ h ⊢ to_tree_order (cast child) →⇩r nodes
⟶ (node ∈ set nodes ⟶ (h ⊢ to_tree_order node →⇩r nodes' ⟶ set nodes' ⊆ set nodes)))"
using 1
by blast
obtain children where children: "h ⊢ get_child_nodes parent →⇩r children"
using 2
by(auto simp add: to_tree_order_def elim!: bind_returns_result_E)
then have "set nodes' ⊆ set nodes"
proof (cases "children = []")
case True
then show ?thesis
by (metis "2" "3" ‹h ⊢ to_tree_order node →⇩r nodes'› children empty_iff list.set(1)
subsetI to_tree_order_either_ptr_or_in_children)
next
case False
then show ?thesis
proof (cases "node = parent")
case True
then show ?thesis
using "2" ‹h ⊢ to_tree_order node →⇩r nodes'› returns_result_eq by fastforce
next
case False
then obtain child nodes_of_child where
"child ∈ set children" and
"h ⊢ to_tree_order (cast child) →⇩r nodes_of_child" and
"node ∈ set nodes_of_child"
using 2[simplified to_tree_order_def] 3
to_tree_order_either_ptr_or_in_children[where node=node and ptr=parent] children
apply(auto elim!: bind_returns_result_E2 intro: map_M_pure_I)[1]
using is_OK_returns_result_E 2 a_all_ptrs_in_heap_def assms(1) heap_is_wellformed_def
using "3" by blast
then have "set nodes' ⊆ set nodes_of_child"
using h1
using ‹h ⊢ to_tree_order node →⇩r nodes'› children by blast
moreover have "set nodes_of_child ⊆ set nodes"
using "2" ‹child ∈ set children› ‹h ⊢ to_tree_order (cast child) →⇩r nodes_of_child›
assms children to_tree_order_child_subset by auto
ultimately show ?thesis
by blast
qed
qed
then show "x ∈ set nodes"
using ‹x ∈ set nodes'› by blast
qed
qed
then show ?thesis
using assms by blast
qed
lemma to_tree_order_parent:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
assumes "h ⊢ get_parent child →⇩r Some parent"
assumes "parent ∈ set nodes"
shows "cast child ∈ set nodes"
proof -
obtain nodes' where nodes': "h ⊢ to_tree_order parent →⇩r nodes'"
using assms to_tree_order_ok get_parent_parent_in_heap
by (meson get_parent_parent_in_heap is_OK_returns_result_E)
then have "set nodes' ⊆ set nodes"
using to_tree_order_subset assms
by blast
moreover obtain children where
children: "h ⊢ get_child_nodes parent →⇩r children" and
child: "child ∈ set children"
using assms get_parent_child_dual by blast
then obtain child_to where child_to: "h ⊢ to_tree_order (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r child_to"
by (meson assms(1) assms(2) assms(3) assms(5) is_OK_returns_result_E is_OK_returns_result_I
get_parent_ptr_in_heap node_ptr_kinds_commutes to_tree_order_ok)
then have "cast child ∈ set child_to"
apply(simp add: to_tree_order_def)
by(auto elim!: bind_returns_result_E2 map_M_pure_E
dest!: bind_returns_result_E3[rotated, OF children, rotated] intro!: map_M_pure_I)
have "cast child ∈ set nodes'"
using nodes' child
apply(simp add: to_tree_order_def)
apply(auto elim!: bind_returns_result_E2 map_M_pure_E
dest!: bind_returns_result_E3[rotated, OF children, rotated] intro!: map_M_pure_I)[1]
using child_to ‹cast child ∈ set child_to› returns_result_eq by fastforce
ultimately show ?thesis
by auto
qed
lemma to_tree_order_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
assumes "h ⊢ get_child_nodes parent →⇩r children"
assumes "cast child ≠ ptr"
assumes "child ∈ set children"
assumes "cast child ∈ set nodes"
shows "parent ∈ set nodes"
proof(insert assms(1) assms(4) assms(6) assms(8), induct ptr arbitrary: nodes
rule: heap_wellformed_induct)
case (step p)
have "p |∈| object_ptr_kinds h"
using ‹h ⊢ to_tree_order p →⇩r nodes› to_tree_order_ptr_in_heap
using assms(1) assms(2) assms(3) by blast
then obtain children where children: "h ⊢ get_child_nodes p →⇩r children"
by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr)
then show ?case
proof (cases "children = []")
case True
then show ?thesis
using step(2) step(3) step(4) children
by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])
next
case False
then obtain c child_to where
child: "c ∈ set children" and
child_to: "h ⊢ to_tree_order (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) →⇩r child_to" and
"cast child ∈ set child_to"
using step(2) children
apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])[1]
by (metis (full_types) assms(1) assms(2) assms(3) get_parent_ptr_in_heap
is_OK_returns_result_I l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M.child_parent_dual
l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms node_ptr_kinds_commutes
returns_result_select_result step.prems(1) step.prems(2) step.prems(3)
to_tree_order_either_ptr_or_in_children to_tree_order_ok)
then have "set child_to ⊆ set nodes"
using assms(1) child children step.prems(1) to_tree_order_child_subset by auto
show ?thesis
proof (cases "c = child")
case True
then have "parent = p"
using step(3) children child assms(5) assms(7)
by (meson assms(1) assms(2) assms(3) child_parent_dual option.inject returns_result_eq)
then show ?thesis
using step.prems(1) to_tree_order_ptr_in_result by blast
next
case False
then show ?thesis
using step(1)[OF children child child_to] step(3) step(4)
using ‹set child_to ⊆ set nodes›
using ‹cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child ∈ set child_to› by auto
qed
qed
qed
lemma to_tree_order_node_ptrs:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
assumes "ptr' ≠ ptr"
assumes "ptr' ∈ set nodes"
shows "is_node_ptr_kind ptr'"
proof(insert assms(1) assms(4) assms(5) assms(6), induct ptr arbitrary: nodes
rule: heap_wellformed_induct)
case (step p)
have "p |∈| object_ptr_kinds h"
using ‹h ⊢ to_tree_order p →⇩r nodes› to_tree_order_ptr_in_heap
using assms(1) assms(2) assms(3) by blast
then obtain children where children: "h ⊢ get_child_nodes p →⇩r children"
by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr)
then show ?case
proof (cases "children = []")
case True
then show ?thesis
using step(2) step(3) step(4) children
by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])[1]
next
case False
then obtain c child_to where
child: "c ∈ set children" and
child_to: "h ⊢ to_tree_order (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) →⇩r child_to" and
"ptr' ∈ set child_to"
using step(2) children
apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])[1]
using step.prems(1) step.prems(2) step.prems(3) to_tree_order_either_ptr_or_in_children by blast
then have "set child_to ⊆ set nodes"
using assms(1) child children step.prems(1) to_tree_order_child_subset by auto
show ?thesis
proof (cases "cast c = ptr")
case True
then show ?thesis
using step ‹ptr' ∈ set child_to› assms(5) child child_to children by blast
next
case False
then show ?thesis
using ‹ptr' ∈ set child_to› child child_to children is_node_ptr_kind_cast step.hyps by blast
qed
qed
qed
lemma to_tree_order_child2:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
assumes "cast child ≠ ptr"
assumes "cast child ∈ set nodes"
obtains parent where "h ⊢ get_parent child →⇩r Some parent" and "parent ∈ set nodes"
proof -
assume 1: "(⋀parent. h ⊢ get_parent child →⇩r Some parent ⟹ parent ∈ set nodes ⟹ thesis)"
show thesis
proof(insert assms(1) assms(4) assms(5) assms(6) 1, induct ptr arbitrary: nodes
rule: heap_wellformed_induct)
case (step p)
have "p |∈| object_ptr_kinds h"
using ‹h ⊢ to_tree_order p →⇩r nodes› to_tree_order_ptr_in_heap
using assms(1) assms(2) assms(3) by blast
then obtain children where children: "h ⊢ get_child_nodes p →⇩r children"
by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr)
then show ?case
proof (cases "children = []")
case True
then show ?thesis
using step(2) step(3) step(4) children
by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])
next
case False
then obtain c child_to where
child: "c ∈ set children" and
child_to: "h ⊢ to_tree_order (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r c) →⇩r child_to" and
"cast child ∈ set child_to"
using step(2) children
apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF children, rotated])[1]
using step.prems(1) step.prems(2) step.prems(3) to_tree_order_either_ptr_or_in_children
by blast
then have "set child_to ⊆ set nodes"
using assms(1) child children step.prems(1) to_tree_order_child_subset by auto
have "cast child |∈| object_ptr_kinds h"
using assms(1) assms(2) assms(3) assms(4) assms(6) to_tree_order_ptrs_in_heap by blast
then obtain parent_opt where parent_opt: "h ⊢ get_parent child →⇩r parent_opt"
by (meson assms(2) assms(3) is_OK_returns_result_E get_parent_ok node_ptr_kinds_commutes)
then show ?thesis
proof (induct parent_opt)
case None
then show ?case
by (metis ‹cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child ∈ set child_to› assms(1) assms(2) assms(3)
cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_inject child child_parent_dual child_to children
option.distinct(1) returns_result_eq step.hyps)
next
case (Some option)
then show ?case
by (meson assms(1) assms(2) assms(3) get_parent_child_dual step.prems(1) step.prems(2)
step.prems(3) step.prems(4) to_tree_order_child)
qed
qed
qed
qed
lemma to_tree_order_parent_child_rel:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r to"
shows "(ptr, child) ∈ (parent_child_rel h)⇧* ⟷ child ∈ set to"
proof
assume 3: "(ptr, child) ∈ (parent_child_rel h)⇧*"
show "child ∈ set to"
proof (insert 3, induct child rule: heap_wellformed_induct_rev[OF assms(1)])
case (1 child)
then show ?case
proof (cases "ptr = child")
case True
then show ?thesis
using assms(4)
apply(simp add: to_tree_order_def)
by(auto simp add: map_M_pure_I elim!: bind_returns_result_E2)
next
case False
obtain child_parent where
"(ptr, child_parent) ∈ (parent_child_rel h)⇧*" and
"(child_parent, child) ∈ (parent_child_rel h)"
using ‹ptr ≠ child›
by (metis "1.prems" rtranclE)
obtain child_node where child_node: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child_node = child"
using ‹(child_parent, child) ∈ parent_child_rel h› node_ptr_casts_commute3
parent_child_rel_node_ptr
by blast
then have "h ⊢ get_parent child_node →⇩r Some child_parent"
using ‹(child_parent, child) ∈ (parent_child_rel h)›
by (meson assms(1) assms(2) assms(3) is_OK_returns_result_E l_get_parent_wf.child_parent_dual
l_heap_is_wellformed.parent_child_rel_child local.get_child_nodes_ok
local.known_ptrs_known_ptr local.l_get_parent_wf_axioms
local.l_heap_is_wellformed_axioms local.parent_child_rel_parent_in_heap)
then show ?thesis
using 1(1) child_node ‹(ptr, child_parent) ∈ (parent_child_rel h)⇧*›
using assms(1) assms(2) assms(3) assms(4) to_tree_order_parent by blast
qed
qed
next
assume "child ∈ set to"
then show "(ptr, child) ∈ (parent_child_rel h)⇧*"
proof (induct child rule: heap_wellformed_induct_rev[OF assms(1)])
case (1 child)
then show ?case
proof (cases "ptr = child")
case True
then show ?thesis
by simp
next
case False
then have "∃parent. (parent, child) ∈ (parent_child_rel h)"
using 1(2) assms(4) to_tree_order_child2[OF assms(1) assms(2) assms(3) assms(4)]
to_tree_order_node_ptrs
by (metis assms(1) assms(2) assms(3) node_ptr_casts_commute3 parent_child_rel_parent)
then obtain child_node where child_node: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child_node = child"
using node_ptr_casts_commute3 parent_child_rel_node_ptr by blast
then obtain child_parent where child_parent: "h ⊢ get_parent child_node →⇩r Some child_parent"
using ‹∃parent. (parent, child) ∈ (parent_child_rel h)›
by (metis "1.prems" False assms(1) assms(2) assms(3) assms(4) to_tree_order_child2)
then have "(child_parent, child) ∈ (parent_child_rel h)"
using assms(1) child_node parent_child_rel_parent by blast
moreover have "child_parent ∈ set to"
by (metis "1.prems" False assms(1) assms(2) assms(3) assms(4) child_node child_parent
get_parent_child_dual to_tree_order_child)
then have "(ptr, child_parent) ∈ (parent_child_rel h)⇧*"
using 1 child_node child_parent by blast
ultimately show ?thesis
by auto
qed
qed
qed
end
interpretation i_to_tree_order_wf?: l_to_tree_order_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes
get_child_nodes_locs to_tree_order known_ptrs get_parent
get_parent_locs heap_is_wellformed parent_child_rel
get_disconnected_nodes get_disconnected_nodes_locs
using instances
apply(simp add: l_to_tree_order_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
done
declare l_to_tree_order_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
locale l_to_tree_order_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs
+ l_to_tree_order_defs
+ l_get_parent_defs + l_get_child_nodes_defs +
assumes to_tree_order_ok:
"heap_is_wellformed h ⟹ ptr |∈| object_ptr_kinds h ⟹ known_ptrs h ⟹ type_wf h
⟹ h ⊢ ok (to_tree_order ptr)"
assumes to_tree_order_ptrs_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r to
⟹ ptr' ∈ set to ⟹ ptr' |∈| object_ptr_kinds h"
assumes to_tree_order_parent_child_rel:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r to
⟹ (ptr, child_ptr) ∈ (parent_child_rel h)⇧* ⟷ child_ptr ∈ set to"
assumes to_tree_order_child2:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r nodes
⟹ cast child ≠ ptr ⟹ cast child ∈ set nodes
⟹ (⋀parent. h ⊢ get_parent child →⇩r Some parent
⟹ parent ∈ set nodes ⟹ thesis)
⟹ thesis"
assumes to_tree_order_node_ptrs:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r nodes
⟹ ptr' ≠ ptr ⟹ ptr' ∈ set nodes ⟹ is_node_ptr_kind ptr'"
assumes to_tree_order_child:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r nodes
⟹ h ⊢ get_child_nodes parent →⇩r children ⟹ cast child ≠ ptr
⟹ child ∈ set children ⟹ cast child ∈ set nodes
⟹ parent ∈ set nodes"
assumes to_tree_order_ptr_in_result:
"h ⊢ to_tree_order ptr →⇩r nodes ⟹ ptr ∈ set nodes"
assumes to_tree_order_parent:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r nodes
⟹ h ⊢ get_parent child →⇩r Some parent ⟹ parent ∈ set nodes
⟹ cast child ∈ set nodes"
assumes to_tree_order_subset:
"heap_is_wellformed h ⟹ h ⊢ to_tree_order ptr →⇩r nodes ⟹ node ∈ set nodes
⟹ h ⊢ to_tree_order node →⇩r nodes' ⟹ known_ptrs h
⟹ type_wf h ⟹ set nodes' ⊆ set nodes"
lemma to_tree_order_wf_is_l_to_tree_order_wf [instances]:
"l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs
to_tree_order get_parent get_child_nodes"
using instances
apply(auto simp add: l_to_tree_order_wf_def l_to_tree_order_wf_axioms_def)[1]
using to_tree_order_ok
apply blast
using to_tree_order_ptrs_in_heap
apply blast
using to_tree_order_parent_child_rel
apply(blast, blast)
using to_tree_order_child2
apply blast
using to_tree_order_node_ptrs
apply blast
using to_tree_order_child
apply blast
using to_tree_order_ptr_in_result
apply blast
using to_tree_order_parent
apply blast
using to_tree_order_subset
apply blast
done
subsubsection ‹get\_root\_node›
locale l_to_tree_order_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
+ l_to_tree_order_wf
begin
lemma to_tree_order_get_root_node:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r to"
assumes "ptr' ∈ set to"
assumes "h ⊢ get_root_node ptr' →⇩r root_ptr"
assumes "ptr'' ∈ set to"
shows "h ⊢ get_root_node ptr'' →⇩r root_ptr"
proof -
obtain ancestors' where ancestors': "h ⊢ get_ancestors ptr' →⇩r ancestors'"
by (meson assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_ok is_OK_returns_result_E
to_tree_order_ptrs_in_heap )
moreover have "ptr ∈ set ancestors'"
using ‹h ⊢ get_ancestors ptr' →⇩r ancestors'›
using assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_parent_child_rel
to_tree_order_parent_child_rel by blast
ultimately have "h ⊢ get_root_node ptr →⇩r root_ptr"
using ‹h ⊢ get_root_node ptr' →⇩r root_ptr›
using assms(1) assms(2) assms(3) get_ancestors_ptr get_ancestors_same_root_node by blast
obtain ancestors'' where ancestors'': "h ⊢ get_ancestors ptr'' →⇩r ancestors''"
by (meson assms(1) assms(2) assms(3) assms(4) assms(7) get_ancestors_ok is_OK_returns_result_E
to_tree_order_ptrs_in_heap)
moreover have "ptr ∈ set ancestors''"
using ‹h ⊢ get_ancestors ptr'' →⇩r ancestors''›
using assms(1) assms(2) assms(3) assms(4) assms(7) get_ancestors_parent_child_rel
to_tree_order_parent_child_rel by blast
ultimately show ?thesis
using ‹h ⊢ get_root_node ptr →⇩r root_ptr› assms(1) assms(2) assms(3) get_ancestors_ptr
get_ancestors_same_root_node by blast
qed
lemma to_tree_order_same_root:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r root_ptr"
assumes "h ⊢ to_tree_order root_ptr →⇩r to"
assumes "ptr' ∈ set to"
shows "h ⊢ get_root_node ptr' →⇩r root_ptr"
proof (insert assms(1) assms(6), induct ptr' rule: heap_wellformed_induct_rev)
case (step child)
then show ?case
proof (cases "h ⊢ get_root_node child →⇩r child")
case True
then have "child = root_ptr"
using assms(1) assms(2) assms(3) assms(5) step.prems
by (metis (no_types, lifting) get_root_node_same_no_parent node_ptr_casts_commute3
option.simps(3) returns_result_eq to_tree_order_child2 to_tree_order_node_ptrs)
then show ?thesis
using True by blast
next
case False
then obtain child_node parent where "cast child_node = child"
and "h ⊢ get_parent child_node →⇩r Some parent"
by (metis assms(1) assms(2) assms(3) assms(4) assms(5) local.get_root_node_no_parent
local.get_root_node_not_node_same local.get_root_node_same_no_parent
local.to_tree_order_child2 local.to_tree_order_ptrs_in_heap node_ptr_casts_commute3
step.prems)
then show ?thesis
proof (cases "child = root_ptr")
case True
then have "h ⊢ get_root_node root_ptr →⇩r root_ptr"
using assms(4)
using ‹cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child_node = child› assms(1) assms(2) assms(3)
get_root_node_no_parent get_root_node_same_no_parent
by blast
then show ?thesis
using step assms(4)
using True by blast
next
case False
then have "parent ∈ set to"
using assms(5) step(2) to_tree_order_child ‹h ⊢ get_parent child_node →⇩r Some parent›
‹cast child_node = child›
by (metis False assms(1) assms(2) assms(3) get_parent_child_dual)
then show ?thesis
using ‹cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child_node = child› ‹h ⊢ get_parent child_node →⇩r Some parent›
get_root_node_parent_same
using step.hyps by blast
qed
qed
qed
end
interpretation i_to_tree_order_wf_get_root_node_wf?: l_to_tree_order_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes
get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_parent get_parent_locs
get_ancestors get_ancestors_locs get_root_node get_root_node_locs to_tree_order
using instances
by(simp add: l_to_tree_order_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
locale l_to_tree_order_wf_get_root_node_wf = l_type_wf + l_known_ptrs + l_to_tree_order_defs
+ l_get_root_node_defs + l_heap_is_wellformed_defs +
assumes to_tree_order_get_root_node:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ to_tree_order ptr →⇩r to
⟹ ptr' ∈ set to ⟹ h ⊢ get_root_node ptr' →⇩r root_ptr
⟹ ptr'' ∈ set to ⟹ h ⊢ get_root_node ptr'' →⇩r root_ptr"
assumes to_tree_order_same_root:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_root_node ptr →⇩r root_ptr
⟹ h ⊢ to_tree_order root_ptr →⇩r to ⟹ ptr' ∈ set to
⟹ h ⊢ get_root_node ptr' →⇩r root_ptr"
lemma to_tree_order_wf_get_root_node_wf_is_l_to_tree_order_wf_get_root_node_wf [instances]:
"l_to_tree_order_wf_get_root_node_wf type_wf known_ptr known_ptrs to_tree_order
get_root_node heap_is_wellformed"
using instances
apply(auto simp add: l_to_tree_order_wf_get_root_node_wf_def
l_to_tree_order_wf_get_root_node_wf_axioms_def)[1]
using to_tree_order_get_root_node apply blast
using to_tree_order_same_root apply blast
done
subsection ‹get\_owner\_document›
locale l_get_owner_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_known_ptrs
+ l_heap_is_wellformed
+ l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
+ l_get_ancestors
+ l_get_ancestors_wf
+ l_get_parent
+ l_get_parent_wf
+ l_get_root_node_wf
+ l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_owner_document_disconnected_nodes:
assumes "heap_is_wellformed h"
assumes "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assumes "node_ptr ∈ set disc_nodes"
assumes known_ptrs: "known_ptrs h"
assumes type_wf: "type_wf h"
shows "h ⊢ get_owner_document (cast node_ptr) →⇩r document_ptr"
proof -
have 2: "node_ptr |∈| node_ptr_kinds h"
using assms heap_is_wellformed_disc_nodes_in_heap
by blast
have 3: "document_ptr |∈| document_ptr_kinds h"
using assms(2) get_disconnected_nodes_ptr_in_heap by blast
have 0:
"∃!document_ptr∈set |h ⊢ document_ptr_kinds_M|⇩r. node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r"
by (metis (no_types, lifting) "3" DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) assms(2) assms(3)
disjoint_iff_not_equal l_heap_is_wellformed.heap_is_wellformed_one_disc_parent
local.get_disconnected_nodes_ok local.l_heap_is_wellformed_axioms
returns_result_select_result select_result_I2 type_wf)
have "h ⊢ get_parent node_ptr →⇩r None"
using heap_is_wellformed_children_disc_nodes_different child_parent_dual assms
using "2" disjoint_iff_not_equal local.get_parent_child_dual local.get_parent_ok
returns_result_select_result split_option_ex
by (metis (no_types, lifting))
then have 4: "h ⊢ get_root_node (cast node_ptr) →⇩r cast node_ptr"
using 2 get_root_node_no_parent
by blast
obtain document_ptrs where document_ptrs: "h ⊢ document_ptr_kinds_M →⇩r document_ptrs"
by simp
then
have "h ⊢ ok (filter_M (λdocument_ptr. do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (((cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)) ∈ cast ` set disconnected_nodes)
}) document_ptrs)"
using assms(1) get_disconnected_nodes_ok type_wf unfolding heap_is_wellformed_def
by(auto intro!: bind_is_OK_I2 filter_M_is_OK_I bind_pure_I)
then obtain candidates where
candidates: "h ⊢ filter_M (λdocument_ptr. do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (((cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)) ∈ cast ` set disconnected_nodes)
}) document_ptrs →⇩r candidates"
by auto
have eq: "⋀document_ptr. document_ptr |∈| document_ptr_kinds h
⟹ node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r ⟷ |h ⊢ do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (((cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)) ∈ cast ` set disconnected_nodes)
}|⇩r"
apply(auto dest!: get_disconnected_nodes_ok[OF type_wf]
intro!: select_result_I[where P=id, simplified] elim!: bind_returns_result_E2)[1]
apply(drule select_result_E[where P=id, simplified])
by(auto elim!: bind_returns_result_E2)
have filter: "filter (λdocument_ptr. |h ⊢ do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr ∈ cast ` set disconnected_nodes)
}|⇩r) document_ptrs = [document_ptr]"
apply(rule filter_ex1)
using 0 document_ptrs apply(simp)[1]
using eq
using local.get_disconnected_nodes_ok apply auto[1]
using assms(2) assms(3)
apply(auto intro!: intro!: select_result_I[where P=id, simplified]
elim!: bind_returns_result_E2)[1]
using returns_result_eq apply fastforce
using document_ptrs 3 apply(simp)
using document_ptrs
by simp
have "h ⊢ filter_M (λdocument_ptr. do {
disconnected_nodes ← get_disconnected_nodes document_ptr;
return (((cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)) ∈ cast ` set disconnected_nodes)
}) document_ptrs →⇩r [document_ptr]"
apply(rule filter_M_filter2)
using get_disconnected_nodes_ok document_ptrs 3 assms(1) type_wf filter
unfolding heap_is_wellformed_def
by(auto intro: bind_pure_I bind_is_OK_I2)
with 4 document_ptrs have "h ⊢ a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr () →⇩r document_ptr"
by(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I
split: option.splits)[1]
moreover have "known_ptr (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr)"
using "4" assms(1) known_ptrs type_wf known_ptrs_known_ptr "2" node_ptr_kinds_commutes by blast
ultimately show ?thesis
using 2
apply(auto simp add: known_ptr_impl get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
by(auto split: option.splits intro!: bind_pure_returns_result_I)
qed
lemma in_disconnected_nodes_no_parent:
assumes "heap_is_wellformed h"
and "h ⊢ get_parent node_ptr →⇩r None"
and "h ⊢ get_owner_document (cast node_ptr) →⇩r owner_document"
and "h ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "node_ptr ∈ set disc_nodes"
proof -
have 2: "cast node_ptr |∈| object_ptr_kinds h"
using assms(3) get_owner_document_ptr_in_heap by fast
then have 3: "h ⊢ get_root_node (cast node_ptr) →⇩r cast node_ptr"
using assms(2) local.get_root_node_no_parent by blast
have "¬(∃parent_ptr. parent_ptr |∈| object_ptr_kinds h ∧
node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)"
apply(auto)[1]
using assms(2) child_parent_dual[OF assms(1)] type_wf
assms(1) assms(5) get_child_nodes_ok known_ptrs_known_ptr option.simps(3)
returns_result_eq returns_result_select_result
by (metis (no_types, hide_lams))
moreover have "node_ptr |∈| node_ptr_kinds h"
using assms(2) get_parent_ptr_in_heap by blast
ultimately
have 0: "∃document_ptr∈set |h ⊢ document_ptr_kinds_M|⇩r. node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r"
by (metis DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) finite_set_in heap_is_wellformed_children_disc_nodes)
then obtain document_ptr where
document_ptr: "document_ptr∈set |h ⊢ document_ptr_kinds_M|⇩r" and
node_ptr_in_disc_nodes: "node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r"
by auto
then show ?thesis
using get_owner_document_disconnected_nodes known_ptrs type_wf assms
using DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) assms(3) assms(4) get_disconnected_nodes_ok
returns_result_select_result select_result_I2
by (metis (no_types, hide_lams) )
qed
lemma get_owner_document_owner_document_in_heap:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_owner_document ptr →⇩r owner_document"
shows "owner_document |∈| document_ptr_kinds h"
using assms
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_split_asm)+
proof -
assume "h ⊢ invoke [] ptr () →⇩r owner_document"
then show "owner_document |∈| document_ptr_kinds h"
by (meson invoke_empty is_OK_returns_result_I)
next
assume "h ⊢ Heap_Error_Monad.bind (check_in_heap ptr)
(λ_. (local.a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ∘ the ∘ cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r) ptr ())
→⇩r owner_document"
then show "owner_document |∈| document_ptr_kinds h"
by(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2 split: if_splits)
next
assume 0: "heap_is_wellformed h"
and 1: "type_wf h"
and 2: "known_ptrs h"
and 3: "¬ is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr"
and 4: "is_character_data_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr"
and 5: "h ⊢ Heap_Error_Monad.bind (check_in_heap ptr)
(λ_. (local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ∘ the ∘ cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r) ptr ()) →⇩r owner_document"
then obtain root where
root: "h ⊢ get_root_node ptr →⇩r root"
by(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
split: option.splits)
then show ?thesis
proof (cases "is_document_ptr root")
case True
then show ?thesis
using 4 5 root
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
apply(drule(1) returns_result_eq) apply(auto)[1]
using "0" "1" "2" document_ptr_kinds_commutes local.get_root_node_root_in_heap by blast
next
case False
have "known_ptr root"
using "0" "1" "2" local.get_root_node_root_in_heap local.known_ptrs_known_ptr root by blast
have "root |∈| object_ptr_kinds h"
using root
using "0" "1" "2" local.get_root_node_root_in_heap
by blast
then have "is_node_ptr_kind root"
using False ‹known_ptr root›
apply(simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs)
using is_node_ptr_kind_none by force
then
have "(∃document_ptr ∈ fset (document_ptr_kinds h).
root ∈ cast ` set |h ⊢ get_disconnected_nodes document_ptr|⇩r)"
by (metis (no_types, lifting) "0" "1" "2" ‹root |∈| object_ptr_kinds h› local.child_parent_dual
local.get_child_nodes_ok local.get_root_node_same_no_parent local.heap_is_wellformed_children_disc_nodes
local.known_ptrs_known_ptr node_ptr_casts_commute3 node_ptr_inclusion node_ptr_kinds_commutes notin_fset
option.distinct(1) returns_result_eq returns_result_select_result root)
then obtain some_owner_document where
"some_owner_document |∈| document_ptr_kinds h" and
"root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r"
by auto
then
obtain candidates where
candidates: "h ⊢ filter_M
(λdocument_ptr.
Heap_Error_Monad.bind (get_disconnected_nodes document_ptr)
(λdisconnected_nodes. return (root ∈ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ` set disconnected_nodes)))
(sorted_list_of_set (fset (document_ptr_kinds h)))
→⇩r candidates"
by (metis (no_types, lifting) "1" bind_is_OK_I2 bind_pure_I filter_M_is_OK_I finite_fset
is_OK_returns_result_E local.get_disconnected_nodes_ok local.get_disconnected_nodes_pure notin_fset
return_ok return_pure sorted_list_of_set(1))
then have "some_owner_document ∈ set candidates"
apply(rule filter_M_in_result_if_ok)
using ‹some_owner_document |∈| document_ptr_kinds h›
‹root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r›
apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
apply (simp add: ‹some_owner_document |∈| document_ptr_kinds h›)
using "1" ‹root ∈ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r›
‹some_owner_document |∈| document_ptr_kinds h›
local.get_disconnected_nodes_ok by auto
then have "candidates ≠ []"
by auto
then have "owner_document ∈ set candidates"
using 5 root 4
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
apply (metis candidates list.set_sel(1) returns_result_eq)
by (metis ‹is_node_ptr_kind root› node_ptr_no_document_ptr_cast returns_result_eq)
then show ?thesis
using candidates
by (meson bind_pure_I bind_returns_result_E2 filter_M_holds_for_result is_OK_returns_result_I
local.get_disconnected_nodes_ptr_in_heap local.get_disconnected_nodes_pure return_pure)
qed
next
assume 0: "heap_is_wellformed h"
and 1: "type_wf h"
and 2: "known_ptrs h"
and 3: "is_element_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr"
and 4: "h ⊢ Heap_Error_Monad.bind (check_in_heap ptr)
(λ_. (local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ∘ the ∘ cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r) ptr ()) →⇩r owner_document"
then obtain root where
root: "h ⊢ get_root_node ptr →⇩r root"
by(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
split: option.splits)
then show ?thesis
proof (cases "is_document_ptr root")
case True
then show ?thesis
using 3 4 root
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
apply(drule(1) returns_result_eq) apply(auto)[1]
using "0" "1" "2" document_ptr_kinds_commutes local.get_root_node_root_in_heap by blast
next
case False
have "known_ptr root"
using "0" "1" "2" local.get_root_node_root_in_heap local.known_ptrs_known_ptr root by blast
have "root |∈| object_ptr_kinds h"
using root
using "0" "1" "2" local.get_root_node_root_in_heap
by blast
then have "is_node_ptr_kind root"
using False ‹known_ptr root›
apply(simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs)
using is_node_ptr_kind_none by force
then
have "(∃document_ptr ∈ fset (document_ptr_kinds h). root ∈
cast ` set |h ⊢ get_disconnected_nodes document_ptr|⇩r)"
by (metis (no_types, lifting) "0" "1" "2" ‹root |∈| object_ptr_kinds h›
local.child_parent_dual local.get_child_nodes_ok local.get_root_node_same_no_parent
local.heap_is_wellformed_children_disc_nodes local.known_ptrs_known_ptr node_ptr_casts_commute3
node_ptr_inclusion node_ptr_kinds_commutes notin_fset option.distinct(1) returns_result_eq
returns_result_select_result root)
then obtain some_owner_document where
"some_owner_document |∈| document_ptr_kinds h" and
"root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r"
by auto
then
obtain candidates where
candidates: "h ⊢ filter_M
(λdocument_ptr.
Heap_Error_Monad.bind (get_disconnected_nodes document_ptr)
(λdisconnected_nodes. return (root ∈ cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ` set disconnected_nodes)))
(sorted_list_of_set (fset (document_ptr_kinds h)))
→⇩r candidates"
by (metis (no_types, lifting) "1" bind_is_OK_I2 bind_pure_I filter_M_is_OK_I finite_fset
is_OK_returns_result_E local.get_disconnected_nodes_ok local.get_disconnected_nodes_pure notin_fset
return_ok return_pure sorted_list_of_set(1))
then have "some_owner_document ∈ set candidates"
apply(rule filter_M_in_result_if_ok)
using ‹some_owner_document |∈| document_ptr_kinds h›
‹root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r›
apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
using ‹some_owner_document |∈| document_ptr_kinds h›
‹root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r›
apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
using ‹some_owner_document |∈| document_ptr_kinds h›
‹root ∈ cast ` set |h ⊢ get_disconnected_nodes some_owner_document|⇩r›
apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
by (simp add: "1" local.get_disconnected_nodes_ok)
then have "candidates ≠ []"
by auto
then have "owner_document ∈ set candidates"
using 4 root 3
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
apply (metis candidates list.set_sel(1) returns_result_eq)
by (metis ‹is_node_ptr_kind root› node_ptr_no_document_ptr_cast returns_result_eq)
then show ?thesis
using candidates
by (meson bind_pure_I bind_returns_result_E2 filter_M_holds_for_result is_OK_returns_result_I
local.get_disconnected_nodes_ptr_in_heap local.get_disconnected_nodes_pure return_pure)
qed
qed
lemma get_owner_document_ok:
assumes "heap_is_wellformed h" "known_ptrs h" "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
shows "h ⊢ ok (get_owner_document ptr)"
proof -
have "known_ptr ptr"
using assms(2) assms(4) local.known_ptrs_known_ptr
by blast
then show ?thesis
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
apply(auto simp add: known_ptr_impl)[1]
using NodeClass.a_known_ptr_def known_ptr_not_character_data_ptr known_ptr_not_document_ptr
known_ptr_not_element_ptr
apply blast
using assms(4)
apply(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_is_OK_pure_I)[1]
apply (metis (no_types, lifting) document_ptr_casts_commute3 document_ptr_kinds_commutes
is_document_ptr_kind_none option.case_eq_if)
using assms(4)
apply(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_is_OK_pure_I)[1]
apply (metis (no_types, lifting) assms(1) assms(2) assms(3) is_node_ptr_kind_none
local.get_root_node_ok node_ptr_casts_commute3 option.case_eq_if)
using assms(4)
apply(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_is_OK_pure_I)[1]
apply(auto split: option.splits
intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I)[1]
using assms(3) local.get_disconnected_nodes_ok
apply blast
apply (simp add: assms(1) assms(2) assms(3) local.get_root_node_ok)
using assms(4)
apply(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
intro!: bind_is_OK_pure_I)[1]
apply(auto split: option.splits
intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I)[1]
apply (simp add: assms(1) assms(2) assms(3) local.get_root_node_ok)[1]
apply(auto split: option.splits
intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I)[1]
using assms(3) local.get_disconnected_nodes_ok by blast
qed
lemma get_owner_document_child_same:
assumes "heap_is_wellformed h" "known_ptrs h" "type_wf h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "child ∈ set children"
shows "h ⊢ get_owner_document ptr →⇩r owner_document ⟷ h ⊢ get_owner_document (cast child) →⇩r owner_document"
proof -
have "ptr |∈| object_ptr_kinds h"
by (meson assms(4) is_OK_returns_result_I local.get_child_nodes_ptr_in_heap)
then have "known_ptr ptr"
using assms(2) local.known_ptrs_known_ptr by blast
have "cast child |∈| object_ptr_kinds h"
using assms(1) assms(4) assms(5) local.heap_is_wellformed_children_in_heap node_ptr_kinds_commutes
by blast
then
have "known_ptr (cast child)"
using assms(2) local.known_ptrs_known_ptr by blast
obtain root where root: "h ⊢ get_root_node ptr →⇩r root"
by (meson ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) is_OK_returns_result_E
local.get_root_node_ok)
then have "h ⊢ get_root_node (cast child) →⇩r root"
using assms(1) assms(2) assms(3) assms(4) assms(5) local.child_parent_dual
local.get_root_node_parent_same
by blast
have "h ⊢ get_owner_document ptr →⇩r owner_document ⟷ h ⊢ a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child () →⇩r owner_document"
proof (cases "is_document_ptr ptr")
case True
then obtain document_ptr where document_ptr: "cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr = ptr"
using case_optionE document_ptr_casts_commute by blast
then have "root = cast document_ptr"
using root
by(auto simp add: get_root_node_def get_ancestors_def elim!: bind_returns_result_E2
split: option.splits)
then have "h ⊢ a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r document_ptr () →⇩r owner_document ⟷
h ⊢ a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child () →⇩r owner_document"
using document_ptr
‹h ⊢ get_root_node (cast child) →⇩r root›[simplified ‹root = cast document_ptr› document_ptr]
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def
elim!: bind_returns_result_E2
dest!: bind_returns_result_E3[rotated, OF ‹h ⊢ get_root_node (cast child) →⇩r root›
[simplified ‹root = cast document_ptr› document_ptr], rotated]
intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I
split: if_splits option.splits)[1]
using ‹ptr |∈| object_ptr_kinds h› document_ptr_kinds_commutes
by blast
then show ?thesis
using ‹known_ptr ptr›
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_impl)[1]
apply(split invoke_splits, ((rule conjI | rule impI)+)?)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
using ‹ptr |∈| object_ptr_kinds h› True
by(auto simp add: document_ptr[symmetric]
intro!: bind_pure_returns_result_I
split: option.splits)
next
case False
then obtain node_ptr where node_ptr: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr = ptr"
using ‹known_ptr ptr›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)
then have "h ⊢ a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr () →⇩r owner_document ⟷
h ⊢ a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child () →⇩r owner_document"
using root ‹h ⊢ get_root_node (cast child) →⇩r root›
unfolding a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def
by (meson bind_pure_returns_result_I bind_returns_result_E3 local.get_root_node_pure)
then show ?thesis
using ‹known_ptr ptr›
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_impl)[1]
apply(split invoke_splits, ((rule conjI | rule impI)+)?)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h› False
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h› False
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h› False
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h› False
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
apply(split invoke_splits, ((rule conjI | rule impI)+)?)+
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
apply (meson invoke_empty is_OK_returns_result_I)
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
apply(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
by(auto simp add: node_ptr[symmetric] intro!: bind_pure_returns_result_I split: )[1]
qed
then show ?thesis
using ‹known_ptr (cast child)›
apply(auto simp add: get_owner_document_def[of "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child"]
a_get_owner_document_tups_def known_ptr_impl)[1]
apply(split invoke_splits, ((rule conjI | rule impI)+)?)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h›
apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h›
apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h›
apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1]
using ‹cast child |∈| object_ptr_kinds h› ‹ptr |∈| object_ptr_kinds h›
apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1]
by (smt ‹cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child |∈| object_ptr_kinds h› cast_document_ptr_not_node_ptr(1)
comp_apply invoke_empty invoke_not invoke_returns_result is_OK_returns_result_I
node_ptr_casts_commute2 option.sel)
qed
end
locale l_get_owner_document_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs
+ l_get_disconnected_nodes_defs + l_get_owner_document_defs
+ l_get_parent_defs +
assumes get_owner_document_disconnected_nodes:
"heap_is_wellformed h ⟹
known_ptrs h ⟹
type_wf h ⟹
h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes ⟹
node_ptr ∈ set disc_nodes ⟹
h ⊢ get_owner_document (cast node_ptr) →⇩r document_ptr"
assumes in_disconnected_nodes_no_parent:
"heap_is_wellformed h ⟹
h ⊢ get_parent node_ptr →⇩r None⟹
h ⊢ get_owner_document (cast node_ptr) →⇩r owner_document ⟹
h ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes ⟹
known_ptrs h ⟹
type_wf h⟹
node_ptr ∈ set disc_nodes"
assumes get_owner_document_owner_document_in_heap:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹
h ⊢ get_owner_document ptr →⇩r owner_document ⟹
owner_document |∈| document_ptr_kinds h"
assumes get_owner_document_ok:
"heap_is_wellformed h ⟹ known_ptrs h ⟹ type_wf h ⟹ ptr |∈| object_ptr_kinds h
⟹ h ⊢ ok (get_owner_document ptr)"
interpretation i_get_owner_document_wf?: l_get_owner_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr known_ptrs type_wf heap_is_wellformed parent_child_rel get_child_nodes
get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_parent get_parent_locs
get_ancestors get_ancestors_locs get_root_node get_root_node_locs get_owner_document
by(auto simp add: l_get_owner_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_owner_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
lemma get_owner_document_wf_is_l_get_owner_document_wf [instances]:
"l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes
get_owner_document get_parent"
using known_ptrs_is_l_known_ptrs
apply(auto simp add: l_get_owner_document_wf_def l_get_owner_document_wf_axioms_def)[1]
using get_owner_document_disconnected_nodes apply fast
using in_disconnected_nodes_no_parent apply fast
using get_owner_document_owner_document_in_heap apply fast
using get_owner_document_ok apply fast
done
subsubsection ‹get\_root\_node›
locale l_get_owner_document_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_root_node_wf +
l_get_owner_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_owner_document_wf
begin
lemma get_root_node_document:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r root"
assumes "is_document_ptr_kind root"
shows "h ⊢ get_owner_document ptr →⇩r the (cast root)"
proof -
have "ptr |∈| object_ptr_kinds h"
using assms(4)
by (meson is_OK_returns_result_I local.get_root_node_ptr_in_heap)
then have "known_ptr ptr"
using assms(3) local.known_ptrs_known_ptr by blast
{
assume "is_document_ptr_kind ptr"
then have "ptr = root"
using assms(4)
by(auto simp add: get_root_node_def get_ancestors_def elim!: bind_returns_result_E2
split: option.splits)
then have ?thesis
using ‹is_document_ptr_kind ptr› ‹known_ptr ptr› ‹ptr |∈| object_ptr_kinds h›
apply(auto simp add: known_ptr_impl get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
by(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def intro!: bind_pure_returns_result_I
split: option.splits)
}
moreover
{
assume "is_node_ptr_kind ptr"
then have ?thesis
using ‹known_ptr ptr› ‹ptr |∈| object_ptr_kinds h›
apply(auto simp add: known_ptr_impl get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl])
apply(drule(1) known_ptr_not_character_data_ptr)
apply(drule(1) known_ptr_not_element_ptr)
apply(simp add: NodeClass.known_ptr_defs)
apply(auto split: option.splits)[1]
using ‹h ⊢ get_root_node ptr →⇩r root› assms(5)
by(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def is_document_ptr_kind_def
intro!: bind_pure_returns_result_I
split: option.splits)[2]
}
ultimately
show ?thesis
using ‹known_ptr ptr›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)
qed
lemma get_root_node_same_owner_document:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r root"
shows "h ⊢ get_owner_document ptr →⇩r owner_document ⟷ h ⊢ get_owner_document root →⇩r owner_document"
proof -
have "ptr |∈| object_ptr_kinds h"
by (meson assms(4) is_OK_returns_result_I local.get_root_node_ptr_in_heap)
have "root |∈| object_ptr_kinds h"
using assms(1) assms(2) assms(3) assms(4) local.get_root_node_root_in_heap by blast
have "known_ptr ptr"
using ‹ptr |∈| object_ptr_kinds h› assms(3) local.known_ptrs_known_ptr by blast
have "known_ptr root"
using ‹root |∈| object_ptr_kinds h› assms(3) local.known_ptrs_known_ptr by blast
show ?thesis
proof (cases "is_document_ptr_kind ptr")
case True
then
have "ptr = root"
using assms(4)
apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2)[1]
by (metis document_ptr_casts_commute3 last_ConsL local.get_ancestors_not_node
node_ptr_no_document_ptr_cast)
then show ?thesis
by auto
next
case False
then have "is_node_ptr_kind ptr"
using ‹known_ptr ptr›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)
then obtain node_ptr where node_ptr: "ptr = cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr"
by (metis node_ptr_casts_commute3)
show ?thesis
proof
assume "h ⊢ get_owner_document ptr →⇩r owner_document"
then have "h ⊢ local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr () →⇩r owner_document"
using node_ptr
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits)+
apply (meson invoke_empty is_OK_returns_result_I)
by(auto elim!: bind_returns_result_E2 split: option.splits)
show "h ⊢ get_owner_document root →⇩r owner_document"
proof (cases "is_document_ptr_kind root")
case True
have "is_document_ptr root"
using True ‹known_ptr root›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
have "root = cast owner_document"
using True
by (smt ‹h ⊢ get_owner_document ptr →⇩r owner_document› assms(1) assms(2) assms(3) assms(4)
document_ptr_casts_commute3 get_root_node_document returns_result_eq)
then show ?thesis
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
using ‹is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r root› apply blast
using ‹root |∈| object_ptr_kinds h›
by(auto simp add: a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def is_node_ptr_kind_none)
next
case False
then have "is_node_ptr_kind root"
using ‹known_ptr root›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then obtain root_node_ptr where root_node_ptr: "root = cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r root_node_ptr"
by (metis node_ptr_casts_commute3)
then have "h ⊢ local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r root_node_ptr () →⇩r owner_document"
using ‹h ⊢ local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr () →⇩r owner_document› assms(4)
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I split: option.splits)[1]
apply (metis assms(1) assms(2) assms(3) local.get_root_node_no_parent
local.get_root_node_same_no_parent node_ptr returns_result_eq)
using ‹is_node_ptr_kind root› node_ptr returns_result_eq by fastforce
then show ?thesis
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
using ‹is_node_ptr_kind root› ‹known_ptr root›
apply(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)[1]
apply(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)[1]
using ‹root |∈| object_ptr_kinds h›
by(auto simp add: root_node_ptr)
qed
next
assume "h ⊢ get_owner_document root →⇩r owner_document"
show "h ⊢ get_owner_document ptr →⇩r owner_document"
proof (cases "is_document_ptr_kind root")
case True
have "root = cast owner_document"
using ‹h ⊢ get_owner_document root →⇩r owner_document›
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits)+
apply (meson invoke_empty is_OK_returns_result_I)
apply(auto simp add: True a_get_owner_document⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
split: if_splits)[1]
apply (metis True cast_document_ptr_not_node_ptr(2) is_document_ptr_kind_obtains
is_node_ptr_kind_none node_ptr_casts_commute3 option.case_eq_if)
by (metis True cast_document_ptr_not_node_ptr(1) document_ptr_casts_commute3
is_node_ptr_kind_none node_ptr_casts_commute3 option.case_eq_if)
then show ?thesis
using assms(1) assms(2) assms(3) assms(4) get_root_node_document
by fastforce
next
case False
then have "is_node_ptr_kind root"
using ‹known_ptr root›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs NodeClass.known_ptr_defs
split: option.splits)
then obtain root_node_ptr where root_node_ptr: "root = cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r root_node_ptr"
by (metis node_ptr_casts_commute3)
then have "h ⊢ local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r root_node_ptr () →⇩r owner_document"
using ‹h ⊢ get_owner_document root →⇩r owner_document›
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits)+
apply (meson invoke_empty is_OK_returns_result_I)
by(auto simp add: is_document_ptr_kind_none elim!: bind_returns_result_E2)
then have "h ⊢ local.a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r node_ptr () →⇩r owner_document"
apply(auto simp add: a_get_owner_document⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I split: option.splits)[1]
using assms(1) assms(2) assms(3) assms(4) local.get_root_node_no_parent
local.get_root_node_same_no_parent node_ptr returns_result_eq root_node_ptr
by fastforce+
then show ?thesis
apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def)[1]
apply(split invoke_splits, (rule conjI | rule impI)+)+
using node_ptr ‹known_ptr ptr› ‹ptr |∈| object_ptr_kinds h›
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs
CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs NodeClass.known_ptr_defs
intro!: bind_pure_returns_result_I split: option.splits)
qed
qed
qed
qed
end
interpretation get_owner_document_wf_get_root_node_wf?: l_get_owner_document_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs
get_ancestors get_ancestors_locs get_root_node get_root_node_locs heap_is_wellformed parent_child_rel
get_disconnected_nodes get_disconnected_nodes_locs get_owner_document
by(auto simp add: l_get_owner_document_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_owner_document_wf_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
locale l_get_owner_document_wf_get_root_node_wf = l_heap_is_wellformed_defs + l_type_wf +
l_known_ptrs + l_get_root_node_defs + l_get_owner_document_defs +
assumes get_root_node_document:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ get_root_node ptr →⇩r root ⟹
is_document_ptr_kind root ⟹ h ⊢ get_owner_document ptr →⇩r the (cast root)"
assumes get_root_node_same_owner_document:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ get_root_node ptr →⇩r root ⟹
h ⊢ get_owner_document ptr →⇩r owner_document ⟷ h ⊢ get_owner_document root →⇩r owner_document"
lemma get_owner_document_wf_get_root_node_wf_is_l_get_owner_document_wf_get_root_node_wf [instances]:
"l_get_owner_document_wf_get_root_node_wf heap_is_wellformed type_wf known_ptr known_ptrs
get_root_node get_owner_document"
apply(auto simp add: l_get_owner_document_wf_get_root_node_wf_def
l_get_owner_document_wf_get_root_node_wf_axioms_def instances)[1]
using get_root_node_document apply blast
using get_root_node_same_owner_document apply (blast, blast)
done
subsection ‹Preserving heap-wellformedness›
subsection ‹set\_attribute›
locale l_set_attribute_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_parent_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_set_attribute⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_set_attribute_get_disconnected_nodes +
l_set_attribute_get_child_nodes
begin
lemma set_attribute_preserves_wellformedness:
assumes "heap_is_wellformed h"
and "h ⊢ set_attribute element_ptr k v →⇩h h'"
shows "heap_is_wellformed h'"
thm preserves_wellformedness_writes_needed
apply(rule preserves_wellformedness_writes_needed[OF assms set_attribute_writes])
using set_attribute_get_child_nodes
apply(fast)
using set_attribute_get_disconnected_nodes apply(fast)
by(auto simp add: all_args_def set_attribute_locs_def)
end
subsection ‹remove\_child›
locale l_remove_child_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_remove_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_heap_is_wellformed +
l_set_disconnected_nodes_get_child_nodes
begin
lemma remove_child_removes_parent:
assumes wellformed: "heap_is_wellformed h"
and remove_child: "h ⊢ remove_child ptr child →⇩h h2"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "h2 ⊢ get_parent child →⇩r None"
proof -
obtain children where children: "h ⊢ get_child_nodes ptr →⇩r children"
using remove_child remove_child_def by auto
then have "child ∈ set children"
using remove_child remove_child_def
by(auto elim!: bind_returns_heap_E dest: returns_result_eq split: if_splits)
then have h1: "⋀other_ptr other_children. other_ptr ≠ ptr
⟹ h ⊢ get_child_nodes other_ptr →⇩r other_children ⟹ child ∉ set other_children"
using assms(1) known_ptrs type_wf child_parent_dual
by (meson child_parent_dual children option.inject returns_result_eq)
have known_ptr: "known_ptr ptr"
using known_ptrs
by (meson is_OK_returns_heap_I l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms
remove_child remove_child_ptr_in_heap)
obtain owner_document disc_nodes h' where
owner_document: "h ⊢ get_owner_document (cast child) →⇩r owner_document" and
disc_nodes: "h ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h': "h ⊢ set_disconnected_nodes owner_document (child # disc_nodes) →⇩h h'" and
h2: "h' ⊢ set_child_nodes ptr (remove1 child children) →⇩h h2"
using assms children unfolding remove_child_def
apply(auto split: if_splits elim!: bind_returns_heap_E)[1]
by (metis (full_types) get_child_nodes_pure get_disconnected_nodes_pure
get_owner_document_pure pure_returns_heap_eq returns_result_eq)
have "object_ptr_kinds h = object_ptr_kinds h2"
using remove_child_writes remove_child unfolding remove_child_locs_def
apply(rule writes_small_big)
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by(auto simp add: reflp_def transp_def)
then have "|h ⊢ object_ptr_kinds_M|⇩r = |h2 ⊢ object_ptr_kinds_M|⇩r"
unfolding object_ptr_kinds_M_defs by simp
have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved type_wf
by(auto simp add: reflp_def transp_def)
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF remove_child_writes remove_child] unfolding remove_child_locs_def
using set_disconnected_nodes_types_preserved set_child_nodes_types_preserved type_wf
apply(auto simp add: reflp_def transp_def)[1]
by blast
then obtain children' where children': "h2 ⊢ get_child_nodes ptr →⇩r children'"
using h2 set_child_nodes_get_child_nodes known_ptr
by (metis ‹object_ptr_kinds h = object_ptr_kinds h2› children get_child_nodes_ok
get_child_nodes_ptr_in_heap is_OK_returns_result_E is_OK_returns_result_I)
have "child ∉ set children'"
by (metis (mono_tags, lifting) ‹type_wf h'› children children' distinct_remove1_removeAll h2
known_ptr local.heap_is_wellformed_children_distinct
local.set_child_nodes_get_child_nodes member_remove remove_code(1) select_result_I2
wellformed)
moreover have "⋀other_ptr other_children. other_ptr ≠ ptr
⟹ h' ⊢ get_child_nodes other_ptr →⇩r other_children ⟹ child ∉ set other_children"
proof -
fix other_ptr other_children
assume a1: "other_ptr ≠ ptr" and a3: "h' ⊢ get_child_nodes other_ptr →⇩r other_children"
have "h ⊢ get_child_nodes other_ptr →⇩r other_children"
using get_child_nodes_reads set_disconnected_nodes_writes h' a3
apply(rule reads_writes_separate_backwards)
using set_disconnected_nodes_get_child_nodes by fast
show "child ∉ set other_children"
using ‹h ⊢ get_child_nodes other_ptr →⇩r other_children› a1 h1 by blast
qed
then have "⋀other_ptr other_children. other_ptr ≠ ptr
⟹ h2 ⊢ get_child_nodes other_ptr →⇩r other_children ⟹ child ∉ set other_children"
proof -
fix other_ptr other_children
assume a1: "other_ptr ≠ ptr" and a3: "h2 ⊢ get_child_nodes other_ptr →⇩r other_children"
have "h' ⊢ get_child_nodes other_ptr →⇩r other_children"
using get_child_nodes_reads set_child_nodes_writes h2 a3
apply(rule reads_writes_separate_backwards)
using set_disconnected_nodes_get_child_nodes a1 set_child_nodes_get_child_nodes_different_pointers
by metis
then show "child ∉ set other_children"
using ‹⋀other_ptr other_children. ⟦other_ptr ≠ ptr; h' ⊢ get_child_nodes other_ptr →⇩r other_children⟧
⟹ child ∉ set other_children› a1 by blast
qed
ultimately have ha: "⋀other_ptr other_children. h2 ⊢ get_child_nodes other_ptr →⇩r other_children
⟹ child ∉ set other_children"
by (metis (full_types) children' returns_result_eq)
moreover obtain ptrs where ptrs: "h2 ⊢ object_ptr_kinds_M →⇩r ptrs"
by (simp add: object_ptr_kinds_M_defs)
moreover have "⋀ptr. ptr ∈ set ptrs ⟹ h2 ⊢ ok (get_child_nodes ptr)"
using ‹type_wf h2› ptrs get_child_nodes_ok known_ptr
using ‹object_ptr_kinds h = object_ptr_kinds h2› known_ptrs local.known_ptrs_known_ptr by auto
ultimately show "h2 ⊢ get_parent child →⇩r None"
apply(auto simp add: get_parent_def intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I)[1]
proof -
have "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child |∈| object_ptr_kinds h"
using get_owner_document_ptr_in_heap owner_document by blast
then show "h2 ⊢ check_in_heap (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r ()"
by (simp add: ‹object_ptr_kinds h = object_ptr_kinds h2› check_in_heap_def)
next
show "(⋀other_ptr other_children. h2 ⊢ get_child_nodes other_ptr →⇩r other_children
⟹ child ∉ set other_children) ⟹
ptrs = sorted_list_of_set (fset (object_ptr_kinds h2)) ⟹
(⋀ptr. ptr |∈| object_ptr_kinds h2 ⟹ h2 ⊢ ok get_child_nodes ptr) ⟹
h2 ⊢ filter_M (λptr. Heap_Error_Monad.bind (get_child_nodes ptr)
(λchildren. return (child ∈ set children))) (sorted_list_of_set (fset (object_ptr_kinds h2))) →⇩r []"
by(auto intro!: filter_M_empty_I bind_pure_I)
qed
qed
end
locale l_remove_child_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_remove_child_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma remove_child_parent_child_rel_subset:
assumes "heap_is_wellformed h"
and "h ⊢ remove_child ptr child →⇩h h'"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "parent_child_rel h' ⊆ parent_child_rel h"
proof (standard, safe)
obtain owner_document children_h h2 disconnected_nodes_h where
owner_document: "h ⊢ get_owner_document (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r owner_document" and
children_h: "h ⊢ get_child_nodes ptr →⇩r children_h" and
child_in_children_h: "child ∈ set children_h" and
disconnected_nodes_h: "h ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h" and
h2: "h ⊢ set_disconnected_nodes owner_document (child # disconnected_nodes_h) →⇩h h2" and
h': "h2 ⊢ set_child_nodes ptr (remove1 child children_h) →⇩h h'"
using assms(2)
apply(auto simp add: remove_child_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_child_nodes_pure]
split: if_splits)[1]
using pure_returns_heap_eq by fastforce
have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_eq: "⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
unfolding object_ptr_kinds_M_defs by simp
then have object_ptr_kinds_eq2: "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
using select_result_eq by force
then have node_ptr_kinds_eq2: "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by auto
then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'"
using node_ptr_kinds_M_eq by auto
have document_ptr_kinds_eq2: "|h ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'"
using document_ptr_kinds_M_eq by auto
have children_eq:
"⋀ptr' children. ptr ≠ ptr' ⟹
h ⊢ get_child_nodes ptr' →⇩r children =h' ⊢ get_child_nodes ptr' →⇩r children"
apply(rule reads_writes_preserved[OF get_child_nodes_reads remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers
by fast
then have children_eq2:
"⋀ptr' children. ptr ≠ ptr' ⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq:
"⋀document_ptr disconnected_nodes. document_ptr ≠ owner_document
⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes
= h' ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes"
apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_child_nodes_get_disconnected_nodes set_disconnected_nodes_get_disconnected_nodes_different_pointers
by (metis (no_types, lifting) Un_iff owner_document select_result_I2)
then have disconnected_nodes_eq2:
"⋀document_ptr. document_ptr ≠ owner_document
⟹ |h ⊢ get_disconnected_nodes document_ptr|⇩r = |h' ⊢ get_disconnected_nodes document_ptr|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes ptr →⇩r children_h"
apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes
h2 children_h] )
by (simp add: set_disconnected_nodes_get_child_nodes)
have "known_ptr ptr"
using assms(3)
using children_h get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr by blast
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes
h2]
using set_disconnected_nodes_types_preserved type_wf
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_child_nodes_writes h']
using set_child_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_h': "h' ⊢ get_child_nodes ptr →⇩r remove1 child children_h"
using assms(2) owner_document h2 disconnected_nodes_h children_h
apply(auto simp add: remove_child_def split: if_splits)[1]
apply(drule bind_returns_heap_E3)
apply(auto split: if_splits)[1]
apply(simp)
apply(auto split: if_splits)[1]
apply(drule bind_returns_heap_E3)
apply(auto)[1]
apply(simp)
apply(drule bind_returns_heap_E3)
apply(auto)[1]
apply(simp)
apply(drule bind_returns_heap_E4)
apply(auto)[1]
apply(simp)
using ‹type_wf h2› set_child_nodes_get_child_nodes ‹known_ptr ptr› h'
by blast
fix parent child
assume a1: "(parent, child) ∈ parent_child_rel h'"
then show "(parent, child) ∈ parent_child_rel h"
proof (cases "parent = ptr")
case True
then show ?thesis
using a1 remove_child_removes_parent[OF assms(1) assms(2)] children_h children_h'
get_child_nodes_ptr_in_heap
apply(auto simp add: parent_child_rel_def object_ptr_kinds_eq )[1]
by (metis notin_set_remove1)
next
case False
then show ?thesis
using a1
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq3 children_eq2)
qed
qed
lemma remove_child_heap_is_wellformed_preserved:
assumes "heap_is_wellformed h"
and "h ⊢ remove_child ptr child →⇩h h'"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "type_wf h'" and "known_ptrs h'" and "heap_is_wellformed h'"
proof -
obtain owner_document children_h h2 disconnected_nodes_h where
owner_document: "h ⊢ get_owner_document (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r owner_document" and
children_h: "h ⊢ get_child_nodes ptr →⇩r children_h" and
child_in_children_h: "child ∈ set children_h" and
disconnected_nodes_h: "h ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h" and
h2: "h ⊢ set_disconnected_nodes owner_document (child # disconnected_nodes_h) →⇩h h2" and
h': "h2 ⊢ set_child_nodes ptr (remove1 child children_h) →⇩h h'"
using assms(2)
apply(auto simp add: remove_child_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_child_nodes_pure] split: if_splits)[1]
using pure_returns_heap_eq by fastforce
have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_eq: "⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
unfolding object_ptr_kinds_M_defs by simp
then have object_ptr_kinds_eq2: "|h ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
using select_result_eq by force
then have node_ptr_kinds_eq2: "|h ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by auto
then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'"
using node_ptr_kinds_M_eq by auto
have document_ptr_kinds_eq2: "|h ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'"
using document_ptr_kinds_M_eq by auto
have children_eq:
"⋀ptr' children. ptr ≠ ptr' ⟹
h ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
apply(rule reads_writes_preserved[OF get_child_nodes_reads remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers
by fast
then have children_eq2:
"⋀ptr' children. ptr ≠ ptr' ⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq: "⋀document_ptr disconnected_nodes. document_ptr ≠ owner_document
⟹ h ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes
= h' ⊢ get_disconnected_nodes document_ptr →⇩r disconnected_nodes"
apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads remove_child_writes assms(2)])
unfolding remove_child_locs_def
using set_child_nodes_get_disconnected_nodes set_disconnected_nodes_get_disconnected_nodes_different_pointers
by (metis (no_types, lifting) Un_iff owner_document select_result_I2)
then have disconnected_nodes_eq2:
"⋀document_ptr. document_ptr ≠ owner_document
⟹ |h ⊢ get_disconnected_nodes document_ptr|⇩r = |h' ⊢ get_disconnected_nodes document_ptr|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes ptr →⇩r children_h"
apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes
h2 children_h] )
by (simp add: set_disconnected_nodes_get_child_nodes)
show "known_ptrs h'"
using object_ptr_kinds_eq3 known_ptrs_preserved ‹known_ptrs h› by blast
have "known_ptr ptr"
using assms(3)
using children_h get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr by blast
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF set_disconnected_nodes_writes h2]
using set_disconnected_nodes_types_preserved type_wf
by(auto simp add: reflp_def transp_def)
then show "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_child_nodes_writes h']
using set_child_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_h': "h' ⊢ get_child_nodes ptr →⇩r remove1 child children_h"
using assms(2) owner_document h2 disconnected_nodes_h children_h
apply(auto simp add: remove_child_def split: if_splits)[1]
apply(drule bind_returns_heap_E3)
apply(auto split: if_splits)[1]
apply(simp)
apply(auto split: if_splits)[1]
apply(drule bind_returns_heap_E3)
apply(auto)[1]
apply(simp)
apply(drule bind_returns_heap_E3)
apply(auto)[1]
apply(simp)
apply(drule bind_returns_heap_E4)
apply(auto)[1]
apply simp
using ‹type_wf h2› set_child_nodes_get_child_nodes ‹known_ptr ptr› h'
by blast
have disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r child # disconnected_nodes_h"
using owner_document assms(2) h2 disconnected_nodes_h
apply (auto simp add: remove_child_def split: if_splits)[1]
apply(drule bind_returns_heap_E2)
apply(auto split: if_splits)[1]
apply(simp)
by(auto simp add: local.set_disconnected_nodes_get_disconnected_nodes split: if_splits)
then have disconnected_nodes_h': "h' ⊢ get_disconnected_nodes owner_document →⇩r child # disconnected_nodes_h"
apply(rule reads_writes_separate_forwards[OF get_disconnected_nodes_reads set_child_nodes_writes h'])
by (simp add: set_child_nodes_get_disconnected_nodes)
moreover have "a_acyclic_heap h"
using assms(1) by (simp add: heap_is_wellformed_def)
have "parent_child_rel h' ⊆ parent_child_rel h"
proof (standard, safe)
fix parent child
assume a1: "(parent, child) ∈ parent_child_rel h'"
then show "(parent, child) ∈ parent_child_rel h"
proof (cases "parent = ptr")
case True
then show ?thesis
using a1 remove_child_removes_parent[OF assms(1) assms(2)] children_h children_h'
get_child_nodes_ptr_in_heap
apply(auto simp add: parent_child_rel_def object_ptr_kinds_eq )[1]
by (metis imageI notin_set_remove1)
next
case False
then show ?thesis
using a1
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq3 children_eq2)
qed
qed
then have "a_acyclic_heap h'"
using ‹a_acyclic_heap h› acyclic_heap_def acyclic_subset by blast
moreover have "a_all_ptrs_in_heap h"
using assms(1) by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h'"
apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3 disconnected_nodes_eq)[1]
apply (metis (no_types, lifting) ‹type_wf h'› assms(2) assms(3) local.get_child_nodes_ok
local.known_ptrs_known_ptr local.remove_child_children_subset notin_fset object_ptr_kinds_eq3
returns_result_select_result subset_code(1) type_wf)
apply (metis (no_types, lifting) assms(2) disconnected_nodes_eq2 disconnected_nodes_h
disconnected_nodes_h' document_ptr_kinds_eq3 finite_set_in local.remove_child_child_in_heap
node_ptr_kinds_eq3 select_result_I2 set_ConsD subset_code(1))
done
moreover have "a_owner_document_valid h"
using assms(1) by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
apply(auto simp add: a_owner_document_valid_def object_ptr_kinds_eq3 document_ptr_kinds_eq3
node_ptr_kinds_eq3)[1]
proof -
fix node_ptr
assume 0: "∀node_ptr∈fset (node_ptr_kinds h'). (∃document_ptr. document_ptr |∈| document_ptr_kinds h' ∧
node_ptr ∈ set |h ⊢ get_disconnected_nodes document_ptr|⇩r) ∨
(∃parent_ptr. parent_ptr |∈| object_ptr_kinds h' ∧ node_ptr ∈ set |h ⊢ get_child_nodes parent_ptr|⇩r)"
and 1: "node_ptr |∈| node_ptr_kinds h'"
and 2: "∀parent_ptr. parent_ptr |∈| object_ptr_kinds h' ⟶
node_ptr ∉ set |h' ⊢ get_child_nodes parent_ptr|⇩r"
then show "∃document_ptr. document_ptr |∈| document_ptr_kinds h'
∧ node_ptr ∈ set |h' ⊢ get_disconnected_nodes document_ptr|⇩r"
proof (cases "node_ptr = child")
case True
show ?thesis
apply(rule exI[where x=owner_document])
using children_eq2 disconnected_nodes_eq2 children_h children_h' disconnected_nodes_h' True
by (metis (no_types, lifting) get_disconnected_nodes_ptr_in_heap is_OK_returns_result_I
list.set_intros(1) select_result_I2)
next
case False
then show ?thesis
using 0 1 2 children_eq2 children_h children_h' disconnected_nodes_eq2 disconnected_nodes_h
disconnected_nodes_h'
apply(auto simp add: children_eq2 disconnected_nodes_eq2 dest!: select_result_I2)[1]
by (metis children_eq2 disconnected_nodes_eq2 finite_set_in in_set_remove1 list.set_intros(2))
qed
qed
moreover
{
have h0: "a_distinct_lists h"
using assms(1) by (simp add: heap_is_wellformed_def)
moreover have ha1: "(⋃x∈set |h ⊢ object_ptr_kinds_M|⇩r. set |h ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈set |h ⊢ document_ptr_kinds_M|⇩r. set |h ⊢ get_disconnected_nodes x|⇩r) = {}"
using ‹a_distinct_lists h›
unfolding a_distinct_lists_def
by(auto)
have ha2: "ptr |∈| object_ptr_kinds h"
using children_h get_child_nodes_ptr_in_heap by blast
have ha3: "child ∈ set |h ⊢ get_child_nodes ptr|⇩r"
using child_in_children_h children_h
by(simp)
have child_not_in: "⋀document_ptr. document_ptr |∈| document_ptr_kinds h
⟹ child ∉ set |h ⊢ get_disconnected_nodes document_ptr|⇩r"
using ha1 ha2 ha3
apply(simp)
using IntI by fastforce
moreover have "distinct |h ⊢ object_ptr_kinds_M|⇩r"
apply(rule select_result_I)
by(auto simp add: object_ptr_kinds_M_defs)
moreover have "distinct |h ⊢ document_ptr_kinds_M|⇩r"
apply(rule select_result_I)
by(auto simp add: document_ptr_kinds_M_defs)
ultimately have "a_distinct_lists h'"
proof(simp (no_asm) add: a_distinct_lists_def, safe)
assume 1: "a_distinct_lists h"
and 3: "distinct |h ⊢ object_ptr_kinds_M|⇩r"
assume 1: "a_distinct_lists h"
and 3: "distinct |h ⊢ object_ptr_kinds_M|⇩r"
have 4: "distinct (concat ((map (λptr. |h ⊢ get_child_nodes ptr|⇩r) |h ⊢ object_ptr_kinds_M|⇩r)))"
using 1 by(auto simp add: a_distinct_lists_def)
show "distinct (concat (map (λptr. |h' ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))"
proof(rule distinct_concat_map_I[OF 3[unfolded object_ptr_kinds_eq2], simplified])
fix x
assume 5: "x |∈| object_ptr_kinds h'"
then have 6: "distinct |h ⊢ get_child_nodes x|⇩r"
using 4 distinct_concat_map_E object_ptr_kinds_eq2 by fastforce
obtain children where children: "h ⊢ get_child_nodes x →⇩r children"
and distinct_children: "distinct children"
by (metis "5" "6" type_wf assms(3) get_child_nodes_ok local.known_ptrs_known_ptr
object_ptr_kinds_eq3 select_result_I)
obtain children' where children': "h' ⊢ get_child_nodes x →⇩r children'"
using children children_eq children_h' by fastforce
then have "distinct children'"
proof (cases "ptr = x")
case True
then show ?thesis
using children distinct_children children_h children_h'
by (metis children' distinct_remove1 returns_result_eq)
next
case False
then show ?thesis
using children distinct_children children_eq[OF False]
using children' distinct_lists_children h0
using select_result_I2 by fastforce
qed
then show "distinct |h' ⊢ get_child_nodes x|⇩r"
using children' by(auto simp add: )
next
fix x y
assume 5: "x |∈| object_ptr_kinds h'" and 6: "y |∈| object_ptr_kinds h'" and 7: "x ≠ y"
obtain children_x where children_x: "h ⊢ get_child_nodes x →⇩r children_x"
by (metis "5" type_wf assms(3) get_child_nodes_ok is_OK_returns_result_E
local.known_ptrs_known_ptr object_ptr_kinds_eq3)
obtain children_y where children_y: "h ⊢ get_child_nodes y →⇩r children_y"
by (metis "6" type_wf assms(3) get_child_nodes_ok is_OK_returns_result_E
local.known_ptrs_known_ptr object_ptr_kinds_eq3)
obtain children_x' where children_x': "h' ⊢ get_child_nodes x →⇩r children_x'"
using children_eq children_h' children_x by fastforce
obtain children_y' where children_y': "h' ⊢ get_child_nodes y →⇩r children_y'"
using children_eq children_h' children_y by fastforce
have "distinct (concat (map (λptr. |h ⊢ get_child_nodes ptr|⇩r) |h ⊢ object_ptr_kinds_M|⇩r))"
using h0 by(auto simp add: a_distinct_lists_def)
then have 8: "set children_x ∩ set children_y = {}"
using "7" assms(1) children_x children_y local.heap_is_wellformed_one_parent by blast
have "set children_x' ∩ set children_y' = {}"
proof (cases "ptr = x")
case True
then have "ptr ≠ y"
by(simp add: 7)
have "children_x' = remove1 child children_x"
using children_h children_h' children_x children_x' True returns_result_eq by fastforce
moreover have "children_y' = children_y"
using children_y children_y' children_eq[OF ‹ptr ≠ y›] by auto
ultimately show ?thesis
using 8 set_remove1_subset by fastforce
next
case False
then show ?thesis
proof (cases "ptr = y")
case True
have "children_y' = remove1 child children_y"
using children_h children_h' children_y children_y' True returns_result_eq by fastforce
moreover have "children_x' = children_x"
using children_x children_x' children_eq[OF ‹ptr ≠ x›] by auto
ultimately show ?thesis
using 8 set_remove1_subset by fastforce
next
case False
have "children_x' = children_x"
using children_x children_x' children_eq[OF ‹ptr ≠ x›] by auto
moreover have "children_y' = children_y"
using children_y children_y' children_eq[OF ‹ptr ≠ y›] by auto
ultimately show ?thesis
using 8 by simp
qed
qed
then show "set |h' ⊢ get_child_nodes x|⇩r ∩ set |h' ⊢ get_child_nodes y|⇩r = {}"
using children_x' children_y'
by (metis (no_types, lifting) select_result_I2)
qed
next
assume 2: "distinct |h ⊢ document_ptr_kinds_M|⇩r"
then have 4: "distinct (sorted_list_of_set (fset (document_ptr_kinds h')))"
by simp
have 3: "distinct (concat (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h')))))"
using h0
by(simp add: a_distinct_lists_def document_ptr_kinds_eq3)
show "distinct (concat (map (λdocument_ptr. |h' ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h')))))"
proof(rule distinct_concat_map_I[OF 4[unfolded document_ptr_kinds_eq3]])
fix x
assume 4: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
have 5: "distinct |h ⊢ get_disconnected_nodes x|⇩r"
using distinct_lists_disconnected_nodes[OF h0] 4 get_disconnected_nodes_ok
by (simp add: type_wf document_ptr_kinds_eq3 select_result_I)
show "distinct |h' ⊢ get_disconnected_nodes x|⇩r"
proof (cases "x = owner_document")
case True
have "child ∉ set |h ⊢ get_disconnected_nodes x|⇩r"
using child_not_in document_ptr_kinds_eq2 "4" by fastforce
moreover have "|h' ⊢ get_disconnected_nodes x|⇩r = child # |h ⊢ get_disconnected_nodes x|⇩r"
using disconnected_nodes_h' disconnected_nodes_h unfolding True
by(simp)
ultimately show ?thesis
using 5 unfolding True
by simp
next
case False
show ?thesis
using "5" False disconnected_nodes_eq2 by auto
qed
next
fix x y
assume 4: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
and 5: "y ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))" and "x ≠ y"
obtain disc_nodes_x where disc_nodes_x: "h ⊢ get_disconnected_nodes x →⇩r disc_nodes_x"
using 4 get_disconnected_nodes_ok[OF ‹type_wf h›, of x] document_ptr_kinds_eq2
by auto
obtain disc_nodes_y where disc_nodes_y: "h ⊢ get_disconnected_nodes y →⇩r disc_nodes_y"
using 5 get_disconnected_nodes_ok[OF ‹type_wf h›, of y] document_ptr_kinds_eq2
by auto
obtain disc_nodes_x' where disc_nodes_x': "h' ⊢ get_disconnected_nodes x →⇩r disc_nodes_x'"
using 4 get_disconnected_nodes_ok[OF ‹type_wf h'›, of x] document_ptr_kinds_eq2
by auto
obtain disc_nodes_y' where disc_nodes_y': "h' ⊢ get_disconnected_nodes y →⇩r disc_nodes_y'"
using 5 get_disconnected_nodes_ok[OF ‹type_wf h'›, of y] document_ptr_kinds_eq2
by auto
have "distinct
(concat (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r) |h ⊢ document_ptr_kinds_M|⇩r))"
using h0 by (simp add: a_distinct_lists_def)
then have 6: "set disc_nodes_x ∩ set disc_nodes_y = {}"
using ‹x ≠ y› assms(1) disc_nodes_x disc_nodes_y local.heap_is_wellformed_one_disc_parent
by blast
have "set disc_nodes_x' ∩ set disc_nodes_y' = {}"
proof (cases "x = owner_document")
case True
then have "y ≠ owner_document"
using ‹x ≠ y› by simp
then have "disc_nodes_y' = disc_nodes_y"
using disconnected_nodes_eq[OF ‹y ≠ owner_document›] disc_nodes_y disc_nodes_y'
by auto
have "disc_nodes_x' = child # disc_nodes_x"
using disconnected_nodes_h' disc_nodes_x disc_nodes_x' True disconnected_nodes_h
returns_result_eq
by fastforce
have "child ∉ set disc_nodes_y"
using child_not_in disc_nodes_y 5
using document_ptr_kinds_eq2 by fastforce
then show ?thesis
apply(unfold ‹disc_nodes_x' = child # disc_nodes_x› ‹disc_nodes_y' = disc_nodes_y›)
using 6 by auto
next
case False
then show ?thesis
proof (cases "y = owner_document")
case True
then have "disc_nodes_x' = disc_nodes_x"
using disconnected_nodes_eq[OF ‹x ≠ owner_document›] disc_nodes_x disc_nodes_x'
by auto
have "disc_nodes_y' = child # disc_nodes_y"
using disconnected_nodes_h' disc_nodes_y disc_nodes_y' True disconnected_nodes_h
returns_result_eq
by fastforce
have "child ∉ set disc_nodes_x"
using child_not_in disc_nodes_x 4
using document_ptr_kinds_eq2 by fastforce
then show ?thesis
apply(unfold ‹disc_nodes_y' = child # disc_nodes_y› ‹disc_nodes_x' = disc_nodes_x›)
using 6 by auto
next
case False
have "disc_nodes_x' = disc_nodes_x"
using disconnected_nodes_eq[OF ‹x ≠ owner_document›] disc_nodes_x disc_nodes_x'
by auto
have "disc_nodes_y' = disc_nodes_y"
using disconnected_nodes_eq[OF ‹y ≠ owner_document›] disc_nodes_y disc_nodes_y'
by auto
then show ?thesis
apply(unfold ‹disc_nodes_y' = disc_nodes_y› ‹disc_nodes_x' = disc_nodes_x›)
using 6 by auto
qed
qed
then show "set |h' ⊢ get_disconnected_nodes x|⇩r ∩ set |h' ⊢ get_disconnected_nodes y|⇩r = {}"
using disc_nodes_x' disc_nodes_y' by auto
qed
next
fix x xa xb
assume 1: "xa ∈ fset (object_ptr_kinds h')"
and 2: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 3: "xb ∈ fset (document_ptr_kinds h')"
and 4: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
obtain disc_nodes where disc_nodes: "h ⊢ get_disconnected_nodes xb →⇩r disc_nodes"
using 3 get_disconnected_nodes_ok[OF ‹type_wf h›, of xb] document_ptr_kinds_eq2 by auto
obtain disc_nodes' where disc_nodes': "h' ⊢ get_disconnected_nodes xb →⇩r disc_nodes'"
using 3 get_disconnected_nodes_ok[OF ‹type_wf h'›, of xb] document_ptr_kinds_eq2 by auto
obtain children where children: "h ⊢ get_child_nodes xa →⇩r children"
by (metis "1" type_wf assms(3) finite_set_in get_child_nodes_ok is_OK_returns_result_E
local.known_ptrs_known_ptr object_ptr_kinds_eq3)
obtain children' where children': "h' ⊢ get_child_nodes xa →⇩r children'"
using children children_eq children_h' by fastforce
have "⋀x. x ∈ set |h ⊢ get_child_nodes xa|⇩r ⟹ x ∈ set |h ⊢ get_disconnected_nodes xb|⇩r ⟹ False"
using 1 3
apply(fold ‹ object_ptr_kinds h = object_ptr_kinds h'›)
apply(fold ‹ document_ptr_kinds h = document_ptr_kinds h'›)
using children disc_nodes h0 apply(auto simp add: a_distinct_lists_def)[1]
by (metis (no_types, lifting) h0 local.distinct_lists_no_parent select_result_I2)
then have 5: "⋀x. x ∈ set children ⟹ x ∈ set disc_nodes ⟹ False"
using children disc_nodes by fastforce
have 6: "|h' ⊢ get_child_nodes xa|⇩r = children'"
using children' by (simp add: )
have 7: "|h' ⊢ get_disconnected_nodes xb|⇩r = disc_nodes'"
using disc_nodes' by (simp add: )
have "False"
proof (cases "xa = ptr")
case True
have "distinct children_h"
using children_h distinct_lists_children h0 ‹known_ptr ptr› by blast
have "|h' ⊢ get_child_nodes ptr|⇩r = remove1 child children_h"
using children_h'
by(simp add: )
have "children = children_h"
using True children children_h by auto
show ?thesis
using disc_nodes' children' 5 2 4 children_h ‹distinct children_h› disconnected_nodes_h'
apply(auto simp add: 6 7
‹xa = ptr› ‹|h' ⊢ get_child_nodes ptr|⇩r = remove1 child children_h› ‹children = children_h›)[1]
by (metis (no_types, lifting) disc_nodes disconnected_nodes_eq2 disconnected_nodes_h
select_result_I2 set_ConsD)
next
case False
have "children' = children"
using children' children children_eq[OF False[symmetric]]
by auto
then show ?thesis
proof (cases "xb = owner_document")
case True
then show ?thesis
using disc_nodes disconnected_nodes_h disconnected_nodes_h'
using "2" "4" "5" "6" "7" False ‹children' = children› assms(1) child_in_children_h
child_parent_dual children children_h disc_nodes' get_child_nodes_ptr_in_heap
list.set_cases list.simps(3) option.simps(1) returns_result_eq set_ConsD
by (metis (no_types, hide_lams) assms(3) type_wf)
next
case False
then show ?thesis
using "2" "4" "5" "6" "7" ‹children' = children› disc_nodes disc_nodes'
disconnected_nodes_eq returns_result_eq
by metis
qed
qed
then show "x ∈ {}"
by simp
qed
}
ultimately show "heap_is_wellformed h'"
using heap_is_wellformed_def by blast
qed
lemma remove_heap_is_wellformed_preserved:
assumes "heap_is_wellformed h"
and "h ⊢ remove child →⇩h h'"
and "known_ptrs h"
and type_wf: "type_wf h"
shows "type_wf h'" and "known_ptrs h'" and "heap_is_wellformed h'"
using assms
by(auto simp add: remove_def intro: remove_child_heap_is_wellformed_preserved
elim!: bind_returns_heap_E2 split: option.splits)
lemma remove_child_removes_child:
assumes wellformed: "heap_is_wellformed h"
and remove_child: "h ⊢ remove_child ptr' child →⇩h h'"
and children: "h' ⊢ get_child_nodes ptr →⇩r children"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "child ∉ set children"
proof -
obtain owner_document children_h h2 disconnected_nodes_h where
owner_document: "h ⊢ get_owner_document (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r owner_document" and
children_h: "h ⊢ get_child_nodes ptr' →⇩r children_h" and
child_in_children_h: "child ∈ set children_h" and
disconnected_nodes_h: "h ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h" and
h2: "h ⊢ set_disconnected_nodes owner_document (child # disconnected_nodes_h) →⇩h h2" and
h': "h2 ⊢ set_child_nodes ptr' (remove1 child children_h) →⇩h h'"
using assms(2)
apply(auto simp add: remove_child_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_child_nodes_pure]
split: if_splits)[1]
using pure_returns_heap_eq
by fastforce
have "object_ptr_kinds h = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes remove_child])
unfolding remove_child_locs_def
using set_child_nodes_pointers_preserved set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
moreover have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF remove_child_writes assms(2)]
using set_child_nodes_types_preserved set_disconnected_nodes_types_preserved type_wf
unfolding remove_child_locs_def
apply(auto simp add: reflp_def transp_def)[1]
by blast
ultimately show ?thesis
using remove_child_removes_parent remove_child_heap_is_wellformed_preserved child_parent_dual
by (meson children known_ptrs local.known_ptrs_preserved option.distinct(1) remove_child
returns_result_eq type_wf wellformed)
qed
lemma remove_child_removes_first_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r node_ptr # children"
assumes "h ⊢ remove_child ptr node_ptr →⇩h h'"
shows "h' ⊢ get_child_nodes ptr →⇩r children"
proof -
obtain h2 disc_nodes owner_document where
"h ⊢ get_owner_document (cast node_ptr) →⇩r owner_document" and
"h ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h2: "h ⊢ set_disconnected_nodes owner_document (node_ptr # disc_nodes) →⇩h h2" and
"h2 ⊢ set_child_nodes ptr children →⇩h h'"
using assms(5)
apply(auto simp add: remove_child_def
dest!: bind_returns_heap_E3[rotated, OF assms(4) get_child_nodes_pure, rotated])[1]
by(auto elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated,OF get_owner_document_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])
have "known_ptr ptr"
by (meson assms(3) assms(4) is_OK_returns_result_I get_child_nodes_ptr_in_heap known_ptrs_known_ptr)
moreover have "h2 ⊢ get_child_nodes ptr →⇩r node_ptr # children"
apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes h2 assms(4)])
using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers
by fast
moreover have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h2]
using ‹type_wf h› set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
ultimately show ?thesis
using set_child_nodes_get_child_nodes‹h2 ⊢ set_child_nodes ptr children →⇩h h'›
by fast
qed
lemma remove_removes_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r node_ptr # children"
assumes "h ⊢ remove node_ptr →⇩h h'"
shows "h' ⊢ get_child_nodes ptr →⇩r children"
proof -
have "h ⊢ get_parent node_ptr →⇩r Some ptr"
using child_parent_dual assms by fastforce
show ?thesis
using assms remove_child_removes_first_child
by(auto simp add: remove_def
dest!: bind_returns_heap_E3[rotated, OF ‹h ⊢ get_parent node_ptr →⇩r Some ptr›, rotated]
bind_returns_heap_E3[rotated, OF assms(4) get_child_nodes_pure, rotated])
qed
lemma remove_for_all_empty_children:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "h ⊢ forall_M remove children →⇩h h'"
shows "h' ⊢ get_child_nodes ptr →⇩r []"
using assms
proof(induct children arbitrary: h h')
case Nil
then show ?case
by simp
next
case (Cons a children)
have "h ⊢ get_parent a →⇩r Some ptr"
using child_parent_dual Cons by fastforce
with Cons show ?case
proof(auto elim!: bind_returns_heap_E)[1]
fix h2
assume 0: "(⋀h h'. heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h ⊢ forall_M remove children →⇩h h' ⟹ h' ⊢ get_child_nodes ptr →⇩r [])"
and 1: "heap_is_wellformed h"
and 2: "type_wf h"
and 3: "known_ptrs h"
and 4: "h ⊢ get_child_nodes ptr →⇩r a # children"
and 5: "h ⊢ get_parent a →⇩r Some ptr"
and 7: "h ⊢ remove a →⇩h h2"
and 8: "h2 ⊢ forall_M remove children →⇩h h'"
then have "h2 ⊢ get_child_nodes ptr →⇩r children"
using remove_removes_child by blast
moreover have "heap_is_wellformed h2"
using 7 1 2 3 remove_child_heap_is_wellformed_preserved(3)
by(auto simp add: remove_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
split: option.splits)
moreover have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF remove_writes 7]
using ‹type_wf h› remove_child_types_preserved
by(auto simp add: a_remove_child_locs_def reflp_def transp_def)
moreover have "object_ptr_kinds h = object_ptr_kinds h2"
using 7
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_writes])
using remove_child_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have "known_ptrs h2"
using 3 known_ptrs_preserved by blast
ultimately show "h' ⊢ get_child_nodes ptr →⇩r []"
using 0 8 by fast
qed
qed
end
locale l_remove_child_wf2 = l_type_wf + l_known_ptrs + l_remove_child_defs + l_heap_is_wellformed_defs
+ l_get_child_nodes_defs + l_remove_defs +
assumes remove_child_preserves_type_wf:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ remove_child ptr child →⇩h h'
⟹ type_wf h'"
assumes remove_child_preserves_known_ptrs:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ remove_child ptr child →⇩h h'
⟹ known_ptrs h'"
assumes remove_child_heap_is_wellformed_preserved:
"type_wf h ⟹ known_ptrs h ⟹ heap_is_wellformed h ⟹ h ⊢ remove_child ptr child →⇩h h'
⟹ heap_is_wellformed h'"
assumes remove_preserves_type_wf:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ remove child →⇩h h'
⟹ type_wf h'"
assumes remove_preserves_known_ptrs:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ remove child →⇩h h'
⟹ known_ptrs h'"
assumes remove_heap_is_wellformed_preserved:
"type_wf h ⟹ known_ptrs h ⟹ heap_is_wellformed h ⟹ h ⊢ remove child →⇩h h'
⟹ heap_is_wellformed h'"
assumes remove_child_removes_child:
"heap_is_wellformed h ⟹ h ⊢ remove_child ptr' child →⇩h h' ⟹ h' ⊢ get_child_nodes ptr →⇩r children
⟹ known_ptrs h ⟹ type_wf h
⟹ child ∉ set children"
assumes remove_child_removes_first_child:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_child_nodes ptr →⇩r node_ptr # children
⟹ h ⊢ remove_child ptr node_ptr →⇩h h'
⟹ h' ⊢ get_child_nodes ptr →⇩r children"
assumes remove_removes_child:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_child_nodes ptr →⇩r node_ptr # children
⟹ h ⊢ remove node_ptr →⇩h h' ⟹ h' ⊢ get_child_nodes ptr →⇩r children"
assumes remove_for_all_empty_children:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ get_child_nodes ptr →⇩r children
⟹ h ⊢ forall_M remove children →⇩h h' ⟹ h' ⊢ get_child_nodes ptr →⇩r []"
interpretation i_remove_child_wf2?: l_remove_child_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_child_nodes get_child_nodes_locs
set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document
get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs
heap_is_wellformed parent_child_rel
by unfold_locales
lemma remove_child_wf2_is_l_remove_child_wf2 [instances]:
"l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove"
apply(auto simp add: l_remove_child_wf2_def l_remove_child_wf2_axioms_def instances)[1]
using remove_child_heap_is_wellformed_preserved apply(fast, fast, fast)
using remove_heap_is_wellformed_preserved apply(fast, fast, fast)
using remove_child_removes_child apply fast
using remove_child_removes_first_child apply fast
using remove_removes_child apply fast
using remove_for_all_empty_children apply fast
done
subsection ‹adopt\_node›
locale l_adopt_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent_wf +
l_get_owner_document_wf +
l_remove_child_wf2 +
l_heap_is_wellformed
begin
lemma adopt_node_removes_first_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ adopt_node owner_document node →⇩h h'"
assumes "h ⊢ get_child_nodes ptr' →⇩r node # children"
shows "h' ⊢ get_child_nodes ptr' →⇩r children"
proof -
obtain old_document parent_opt h2 where
old_document: "h ⊢ get_owner_document (cast node) →⇩r old_document" and
parent_opt: "h ⊢ get_parent node →⇩r parent_opt" and
h2: "h ⊢ (case parent_opt of Some parent ⇒ do { remove_child parent node }
| None ⇒ do { return ()}) →⇩h h2" and
h': "h2 ⊢ (if owner_document ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 node old_disc_nodes);
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (node # disc_nodes)
} else do { return () }) →⇩h h'"
using assms(4)
by(auto simp add: adopt_node_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_parent_pure])
have "h2 ⊢ get_child_nodes ptr' →⇩r children"
using h2 remove_child_removes_first_child assms(1) assms(2) assms(3) assms(5)
by (metis list.set_intros(1) local.child_parent_dual option.simps(5) parent_opt returns_result_eq)
then
show ?thesis
using h'
by(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
dest!: reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes]
split: if_splits)
qed
lemma adopt_node_document_in_heap:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ ok (adopt_node owner_document node)"
shows "owner_document |∈| document_ptr_kinds h"
proof -
obtain old_document parent_opt h2 h' where
old_document: "h ⊢ get_owner_document (cast node) →⇩r old_document" and
parent_opt: "h ⊢ get_parent node →⇩r parent_opt" and
h2: "h ⊢ (case parent_opt of Some parent ⇒ do { remove_child parent node } | None ⇒ do { return ()}) →⇩h h2"
and
h': "h2 ⊢ (if owner_document ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 node old_disc_nodes);
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (node # disc_nodes)
} else do { return () }) →⇩h h'"
using assms(4)
by(auto simp add: adopt_node_def
elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_parent_pure])
show ?thesis
proof (cases "owner_document = old_document")
case True
then show ?thesis
using old_document get_owner_document_owner_document_in_heap assms(1) assms(2) assms(3)
by auto
next
case False
then obtain h3 old_disc_nodes disc_nodes where
old_disc_nodes: "h2 ⊢ get_disconnected_nodes old_document →⇩r old_disc_nodes" and
h3: "h2 ⊢ set_disconnected_nodes old_document (remove1 node old_disc_nodes) →⇩h h3" and
old_disc_nodes: "h3 ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h': "h3 ⊢ set_disconnected_nodes owner_document (node # disc_nodes) →⇩h h'"
using h'
by(auto elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
then have "owner_document |∈| document_ptr_kinds h3"
by (meson is_OK_returns_result_I local.get_disconnected_nodes_ptr_in_heap)
moreover have "object_ptr_kinds h = object_ptr_kinds h2"
using h2 apply(simp split: option.splits)
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes])
using remove_child_pointers_preserved
by (auto simp add: reflp_def transp_def)
moreover have "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
ultimately show ?thesis
by(auto simp add: document_ptr_kinds_def)
qed
qed
end
locale l_adopt_node_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_adopt_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_root_node +
l_get_owner_document_wf +
l_remove_child_wf2 +
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma adopt_node_removes_child_step:
assumes wellformed: "heap_is_wellformed h"
and adopt_node: "h ⊢ adopt_node owner_document node_ptr →⇩h h2"
and children: "h2 ⊢ get_child_nodes ptr →⇩r children"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "node_ptr ∉ set children"
proof -
obtain old_document parent_opt h' where
old_document: "h ⊢ get_owner_document (cast node_ptr) →⇩r old_document" and
parent_opt: "h ⊢ get_parent node_ptr →⇩r parent_opt" and
h': "h ⊢ (case parent_opt of Some parent ⇒ remove_child parent node_ptr | None ⇒ return () ) →⇩h h'"
using adopt_node get_parent_pure
by(auto simp add: adopt_node_def
elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
split: if_splits)
then have "h' ⊢ get_child_nodes ptr →⇩r children"
using adopt_node
apply(auto simp add: adopt_node_def
dest!: bind_returns_heap_E3[rotated, OF old_document, rotated]
bind_returns_heap_E3[rotated, OF parent_opt, rotated]
elim!: bind_returns_heap_E4[rotated, OF h', rotated])[1]
apply(auto split: if_splits
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1]
apply (simp add: set_disconnected_nodes_get_child_nodes children
reads_writes_preserved[OF get_child_nodes_reads set_disconnected_nodes_writes])
using children by blast
show ?thesis
proof(insert parent_opt h', induct parent_opt)
case None
then show ?case
using child_parent_dual wellformed known_ptrs type_wf
‹h' ⊢ get_child_nodes ptr →⇩r children› returns_result_eq
by fastforce
next
case (Some option)
then show ?case
using remove_child_removes_child ‹h' ⊢ get_child_nodes ptr →⇩r children› known_ptrs type_wf
wellformed
by auto
qed
qed
lemma adopt_node_removes_child:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ adopt_node owner_document node_ptr →⇩h h'"
shows "⋀ptr' children'.
h' ⊢ get_child_nodes ptr' →⇩r children' ⟹ node_ptr ∉ set children'"
using adopt_node_removes_child_step assms by blast
lemma adopt_node_preserves_wellformedness:
assumes "heap_is_wellformed h"
and "h ⊢ adopt_node document_ptr child →⇩h h'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
proof -
obtain old_document parent_opt h2 where
old_document: "h ⊢ get_owner_document (cast child) →⇩r old_document"
and
parent_opt: "h ⊢ get_parent child →⇩r parent_opt"
and
h2: "h ⊢ (case parent_opt of Some parent ⇒ remove_child parent child | None ⇒ return ()) →⇩h h2"
and
h': "h2 ⊢ (if document_ptr ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 child old_disc_nodes);
disc_nodes ← get_disconnected_nodes document_ptr;
set_disconnected_nodes document_ptr (child # disc_nodes)
} else do {
return ()
}) →⇩h h'"
using assms(2)
by(auto simp add: adopt_node_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_parent_pure])
have object_ptr_kinds_h_eq3: "object_ptr_kinds h = object_ptr_kinds h2"
using h2 apply(simp split: option.splits)
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes])
using remove_child_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h:
"⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h2 ⊢ object_ptr_kinds_M →⇩r ptrs"
unfolding object_ptr_kinds_M_defs by simp
then have object_ptr_kinds_eq_h: "|h ⊢ object_ptr_kinds_M|⇩r = |h2 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq_h: "|h ⊢ node_ptr_kinds_M|⇩r = |h2 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have wellformed_h2: "heap_is_wellformed h2"
using h2 remove_child_heap_is_wellformed_preserved known_ptrs type_wf
by (metis (no_types, lifting) assms(1) option.case_eq_if pure_returns_heap_eq return_pure)
have "type_wf h2"
using h2 remove_child_preserves_type_wf known_ptrs type_wf
by (metis (no_types, lifting) assms(1) option.case_eq_if pure_returns_heap_eq return_pure)
have "known_ptrs h2"
using h2 remove_child_preserves_known_ptrs known_ptrs type_wf
by (metis (no_types, lifting) assms(1) option.case_eq_if pure_returns_heap_eq return_pure)
have "heap_is_wellformed h' ∧ known_ptrs h' ∧ type_wf h'"
proof(cases "document_ptr = old_document")
case True
then show ?thesis
using h' wellformed_h2 ‹type_wf h2› ‹known_ptrs h2› by auto
next
case False
then obtain h3 old_disc_nodes disc_nodes_document_ptr_h3 where
docs_neq: "document_ptr ≠ old_document" and
old_disc_nodes: "h2 ⊢ get_disconnected_nodes old_document →⇩r old_disc_nodes" and
h3: "h2 ⊢ set_disconnected_nodes old_document (remove1 child old_disc_nodes) →⇩h h3" and
disc_nodes_document_ptr_h3:
"h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_document_ptr_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (child # disc_nodes_document_ptr_h3) →⇩h h'"
using h'
by(auto elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
have object_ptr_kinds_h2_eq3: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h2:
"⋀ptrs. h2 ⊢ object_ptr_kinds_M →⇩r ptrs = h3 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_eq_h2: "|h2 ⊢ object_ptr_kinds_M|⇩r = |h3 ⊢ object_ptr_kinds_M|⇩r"
by(simp)
then have node_ptr_kinds_eq_h2: "|h2 ⊢ node_ptr_kinds_M|⇩r = |h3 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
then have node_ptr_kinds_eq3_h2: "node_ptr_kinds h2 = node_ptr_kinds h3"
by auto
have document_ptr_kinds_eq2_h2: "|h2 ⊢ document_ptr_kinds_M|⇩r = |h3 ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3_h2: "document_ptr_kinds h2 = document_ptr_kinds h3"
using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
have children_eq_h2:
"⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children = h3 ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h2: "⋀ptr. |h2 ⊢ get_child_nodes ptr|⇩r = |h3 ⊢ get_child_nodes ptr|⇩r"
using select_result_eq by force
have object_ptr_kinds_h3_eq3: "object_ptr_kinds h3 = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h3:
"⋀ptrs. h3 ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_eq_h3: "|h3 ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by(simp)
then have node_ptr_kinds_eq_h3: "|h3 ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
then have node_ptr_kinds_eq3_h3: "node_ptr_kinds h3 = node_ptr_kinds h'"
by auto
have document_ptr_kinds_eq2_h3: "|h3 ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3_h3: "document_ptr_kinds h3 = document_ptr_kinds h'"
using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto
have children_eq_h3:
"⋀ptr children. h3 ⊢ get_child_nodes ptr →⇩r children = h' ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by (simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h3: "⋀ptr. |h3 ⊢ get_child_nodes ptr|⇩r = |h' ⊢ get_child_nodes ptr|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. old_document ≠ doc_ptr
⟹ h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes = h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. old_document ≠ doc_ptr
⟹ |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
obtain disc_nodes_old_document_h2 where disc_nodes_old_document_h2:
"h2 ⊢ get_disconnected_nodes old_document →⇩r disc_nodes_old_document_h2"
using old_disc_nodes by blast
then have disc_nodes_old_document_h3:
"h3 ⊢ get_disconnected_nodes old_document →⇩r remove1 child disc_nodes_old_document_h2"
using h3 old_disc_nodes returns_result_eq set_disconnected_nodes_get_disconnected_nodes
by fastforce
have "distinct disc_nodes_old_document_h2"
using disc_nodes_old_document_h2 local.heap_is_wellformed_disconnected_nodes_distinct wellformed_h2
by blast
have "type_wf h2"
proof (insert h2, induct parent_opt)
case None
then show ?case
using type_wf by simp
next
case (Some option)
then show ?case
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF remove_child_writes]
type_wf remove_child_types_preserved
by (simp add: reflp_def transp_def)
qed
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have "known_ptrs h3"
using known_ptrs local.known_ptrs_preserved object_ptr_kinds_h2_eq3 object_ptr_kinds_h_eq3
by blast
then have "known_ptrs h'"
using local.known_ptrs_preserved object_ptr_kinds_h3_eq3 by blast
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. document_ptr ≠ doc_ptr
⟹ h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes = h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by (simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disc_nodes_document_ptr_h2:
"h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_document_ptr_h3"
using disconnected_nodes_eq_h2 docs_neq disc_nodes_document_ptr_h3 by auto
have disc_nodes_document_ptr_h': "
h' ⊢ get_disconnected_nodes document_ptr →⇩r child # disc_nodes_document_ptr_h3"
using h' disc_nodes_document_ptr_h3
using set_disconnected_nodes_get_disconnected_nodes by blast
have document_ptr_in_heap: "document_ptr |∈| document_ptr_kinds h2"
using disc_nodes_document_ptr_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1)
unfolding heap_is_wellformed_def
using disc_nodes_document_ptr_h2 get_disconnected_nodes_ptr_in_heap by blast
have old_document_in_heap: "old_document |∈| document_ptr_kinds h2"
using disc_nodes_old_document_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1)
unfolding heap_is_wellformed_def
using get_disconnected_nodes_ptr_in_heap old_disc_nodes by blast
have "child ∈ set disc_nodes_old_document_h2"
proof (insert parent_opt h2, induct parent_opt)
case None
then have "h = h2"
by(auto)
moreover have "a_owner_document_valid h"
using assms(1) heap_is_wellformed_def by(simp add: heap_is_wellformed_def)
ultimately show ?case
using old_document disc_nodes_old_document_h2 None(1) child_parent_dual[OF assms(1)]
in_disconnected_nodes_no_parent assms(1) known_ptrs type_wf by blast
next
case (Some option)
then show ?case
apply(simp split: option.splits)
using assms(1) disc_nodes_old_document_h2 old_document remove_child_in_disconnected_nodes
known_ptrs
by blast
qed
have "child ∉ set (remove1 child disc_nodes_old_document_h2)"
using disc_nodes_old_document_h3 h3 known_ptrs wellformed_h2 ‹distinct disc_nodes_old_document_h2›
by auto
have "child ∉ set disc_nodes_document_ptr_h3"
proof -
have "a_distinct_lists h2"
using heap_is_wellformed_def wellformed_h2 by blast
then have 0: "distinct (concat (map (λdocument_ptr. |h2 ⊢ get_disconnected_nodes document_ptr|⇩r)
|h2 ⊢ document_ptr_kinds_M|⇩r))"
by(simp add: a_distinct_lists_def)
show ?thesis
using distinct_concat_map_E(1)[OF 0] ‹child ∈ set disc_nodes_old_document_h2›
disc_nodes_old_document_h2 disc_nodes_document_ptr_h2
by (meson ‹type_wf h2› docs_neq known_ptrs local.get_owner_document_disconnected_nodes
local.known_ptrs_preserved object_ptr_kinds_h_eq3 returns_result_eq wellformed_h2)
qed
have child_in_heap: "child |∈| node_ptr_kinds h"
using get_owner_document_ptr_in_heap[OF is_OK_returns_result_I[OF old_document]]
node_ptr_kinds_commutes by blast
have "a_acyclic_heap h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
have "parent_child_rel h' ⊆ parent_child_rel h2"
proof
fix x
assume "x ∈ parent_child_rel h'"
then show "x ∈ parent_child_rel h2"
using object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3 children_eq2_h2 children_eq2_h3
mem_Collect_eq object_ptr_kinds_M_eq_h3 select_result_eq split_cong
unfolding parent_child_rel_def
by(simp)
qed
then have "a_acyclic_heap h'"
using ‹a_acyclic_heap h2› acyclic_heap_def acyclic_subset by blast
moreover have "a_all_ptrs_in_heap h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h3"
apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h2 children_eq_h2)[1]
apply (simp add: children_eq2_h2 object_ptr_kinds_h2_eq3 subset_code(1))
by (metis (no_types, lifting) ‹child ∈ set disc_nodes_old_document_h2› ‹type_wf h2›
disc_nodes_old_document_h2 disc_nodes_old_document_h3 disconnected_nodes_eq2_h2
document_ptr_kinds_eq3_h2 in_set_remove1 local.get_disconnected_nodes_ok
local.heap_is_wellformed_disc_nodes_in_heap node_ptr_kinds_eq3_h2 returns_result_select_result
select_result_I2 wellformed_h2)
then have "a_all_ptrs_in_heap h'"
apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h3 children_eq_h3)[1]
apply (simp add: children_eq2_h3 object_ptr_kinds_h3_eq3 subset_code(1))
by (metis (no_types, lifting) ‹child ∈ set disc_nodes_old_document_h2›
disc_nodes_document_ptr_h' disc_nodes_document_ptr_h2 disc_nodes_old_document_h2
disconnected_nodes_eq2_h3 document_ptr_kinds_eq3_h3 finite_set_in
local.heap_is_wellformed_disc_nodes_in_heap node_ptr_kinds_eq3_h2 node_ptr_kinds_eq3_h3
select_result_I2 set_ConsD subset_code(1) wellformed_h2)
moreover have "a_owner_document_valid h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
apply(simp add: a_owner_document_valid_def node_ptr_kinds_eq_h2 node_ptr_kinds_eq3_h3
object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 document_ptr_kinds_eq2_h2
document_ptr_kinds_eq2_h3 children_eq2_h2 children_eq2_h3 )
by (smt disc_nodes_document_ptr_h' disc_nodes_document_ptr_h2
disc_nodes_old_document_h2 disc_nodes_old_document_h3
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_in_heap
document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3 in_set_remove1
list.set_intros(1) list.set_intros(2) node_ptr_kinds_eq3_h2
node_ptr_kinds_eq3_h3 object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3
select_result_I2)
have a_distinct_lists_h2: "a_distinct_lists h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h'"
apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h3 object_ptr_kinds_eq_h2
children_eq2_h2 children_eq2_h3)[1]
proof -
assume 1: "distinct (concat (map (λptr. |h' ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))"
and 2: "distinct (concat (map (λdocument_ptr. |h2 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h2)))))"
and 3: "(⋃x∈fset (object_ptr_kinds h'). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h2). set |h2 ⊢ get_disconnected_nodes x|⇩r) = {}"
show "distinct (concat (map (λdocument_ptr. |h' ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h')))))"
proof(rule distinct_concat_map_I)
show "distinct (sorted_list_of_set (fset (document_ptr_kinds h')))"
by(auto simp add: document_ptr_kinds_M_def )
next
fix x
assume a1: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
have 4: "distinct |h2 ⊢ get_disconnected_nodes x|⇩r"
using a_distinct_lists_h2 "2" a1 concat_map_all_distinct document_ptr_kinds_eq2_h2
document_ptr_kinds_eq2_h3
by fastforce
then show "distinct |h' ⊢ get_disconnected_nodes x|⇩r"
proof (cases "old_document ≠ x")
case True
then show ?thesis
proof (cases "document_ptr ≠ x")
case True
then show ?thesis
using disconnected_nodes_eq2_h2[OF ‹old_document ≠ x›]
disconnected_nodes_eq2_h3[OF ‹document_ptr ≠ x›] 4
by(auto)
next
case False
then show ?thesis
using disc_nodes_document_ptr_h3 disc_nodes_document_ptr_h' 4
‹child ∉ set disc_nodes_document_ptr_h3›
by(auto simp add: disconnected_nodes_eq2_h2[OF ‹old_document ≠ x›] )
qed
next
case False
then show ?thesis
by (metis (no_types, hide_lams) ‹distinct disc_nodes_old_document_h2›
disc_nodes_old_document_h3 disconnected_nodes_eq2_h3
distinct_remove1 docs_neq select_result_I2)
qed
next
fix x y
assume a0: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
and a1: "y ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
and a2: "x ≠ y"
moreover have 5: "set |h2 ⊢ get_disconnected_nodes x|⇩r ∩ set |h2 ⊢ get_disconnected_nodes y|⇩r = {}"
using 2 calculation
by (auto simp add: document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3 dest: distinct_concat_map_E(1))
ultimately show "set |h' ⊢ get_disconnected_nodes x|⇩r ∩ set |h' ⊢ get_disconnected_nodes y|⇩r = {}"
proof(cases "old_document = x")
case True
have "old_document ≠ y"
using ‹x ≠ y› ‹old_document = x› by simp
have "document_ptr ≠ x"
using docs_neq ‹old_document = x› by auto
show ?thesis
proof(cases "document_ptr = y")
case True
then show ?thesis
using 5 True select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_document_ptr_h2]
select_result_I2[OF disc_nodes_old_document_h2]
select_result_I2[OF disc_nodes_old_document_h3] ‹old_document = x›
by (metis (no_types, lifting) ‹child ∉ set (remove1 child disc_nodes_old_document_h2)›
‹document_ptr ≠ x› disconnected_nodes_eq2_h3 disjoint_iff_not_equal
notin_set_remove1 set_ConsD)
next
case False
then show ?thesis
using 5 select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_document_ptr_h2]
select_result_I2[OF disc_nodes_old_document_h2]
select_result_I2[OF disc_nodes_old_document_h3]
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 ‹old_document = x›
docs_neq ‹old_document ≠ y›
by (metis (no_types, lifting) disjoint_iff_not_equal notin_set_remove1)
qed
next
case False
then show ?thesis
proof(cases "old_document = y")
case True
then show ?thesis
proof(cases "document_ptr = x")
case True
show ?thesis
using 5 select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_document_ptr_h2]
select_result_I2[OF disc_nodes_old_document_h2]
select_result_I2[OF disc_nodes_old_document_h3]
‹old_document ≠ x› ‹old_document = y› ‹document_ptr = x›
apply(simp)
by (metis (no_types, lifting) ‹child ∉ set (remove1 child disc_nodes_old_document_h2)›
disconnected_nodes_eq2_h3 disjoint_iff_not_equal notin_set_remove1)
next
case False
then show ?thesis
using 5 select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_document_ptr_h2]
select_result_I2[OF disc_nodes_old_document_h2]
select_result_I2[OF disc_nodes_old_document_h3]
‹old_document ≠ x› ‹old_document = y› ‹document_ptr ≠ x›
by (metis (no_types, lifting) disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3
disjoint_iff_not_equal docs_neq notin_set_remove1)
qed
next
case False
have "set |h2 ⊢ get_disconnected_nodes y|⇩r ∩ set disc_nodes_old_document_h2 = {}"
by (metis DocumentMonad.ptr_kinds_M_ok DocumentMonad.ptr_kinds_M_ptr_kinds False
‹type_wf h2› a1 disc_nodes_old_document_h2 document_ptr_kinds_M_def
document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3
l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok
local.heap_is_wellformed_one_disc_parent returns_result_select_result
wellformed_h2)
then show ?thesis
proof(cases "document_ptr = x")
case True
then have "document_ptr ≠ y"
using ‹x ≠ y› by auto
have "set |h2 ⊢ get_disconnected_nodes y|⇩r ∩ set disc_nodes_old_document_h2 = {}"
using ‹set |h2 ⊢ get_disconnected_nodes y|⇩r ∩ set disc_nodes_old_document_h2 = {}›
by blast
then show ?thesis
using 5 select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_document_ptr_h2]
select_result_I2[OF disc_nodes_old_document_h2]
select_result_I2[OF disc_nodes_old_document_h3]
‹old_document ≠ x› ‹old_document ≠ y› ‹document_ptr = x› ‹document_ptr ≠ y›
‹child ∈ set disc_nodes_old_document_h2› disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3
‹set |h2 ⊢ get_disconnected_nodes y|⇩r ∩ set disc_nodes_old_document_h2 = {}›
by(auto)
next
case False
then show ?thesis
proof(cases "document_ptr = y")
case True
have f1: "set |h2 ⊢ get_disconnected_nodes x|⇩r ∩ set disc_nodes_document_ptr_h3 = {}"
using 2 a1 document_ptr_in_heap document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3
‹document_ptr ≠ x› select_result_I2[OF disc_nodes_document_ptr_h3, symmetric]
disconnected_nodes_eq2_h2[OF docs_neq[symmetric], symmetric]
by (simp add: "5" True)
moreover have f1:
"set |h2 ⊢ get_disconnected_nodes x|⇩r ∩ set |h2 ⊢ get_disconnected_nodes old_document|⇩r = {}"
using 2 a1 old_document_in_heap document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3
‹old_document ≠ x›
by (metis (no_types, lifting) a0 distinct_concat_map_E(1) document_ptr_kinds_eq3_h2
document_ptr_kinds_eq3_h3 finite_fset fmember.rep_eq set_sorted_list_of_set)
ultimately show ?thesis
using 5 select_result_I2[OF disc_nodes_document_ptr_h']
select_result_I2[OF disc_nodes_old_document_h2] ‹old_document ≠ x›
‹document_ptr ≠ x› ‹document_ptr = y›
‹child ∈ set disc_nodes_old_document_h2› disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3
by auto
next
case False
then show ?thesis
using 5
select_result_I2[OF disc_nodes_old_document_h2] ‹old_document ≠ x›
‹document_ptr ≠ x› ‹document_ptr ≠ y›
‹child ∈ set disc_nodes_old_document_h2›
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3
by (metis ‹set |h2 ⊢ get_disconnected_nodes y|⇩r ∩ set disc_nodes_old_document_h2 = {}›
empty_iff inf.idem)
qed
qed
qed
qed
qed
next
fix x xa xb
assume 0: "distinct (concat (map (λptr. |h' ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))"
and 1: "distinct (concat (map (λdocument_ptr. |h2 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h2)))))"
and 2: "(⋃x∈fset (object_ptr_kinds h'). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h2). set |h2 ⊢ get_disconnected_nodes x|⇩r) = {}"
and 3: "xa |∈| object_ptr_kinds h'"
and 4: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 5: "xb |∈| document_ptr_kinds h'"
and 6: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
then show False
using ‹child ∈ set disc_nodes_old_document_h2› disc_nodes_document_ptr_h'
disc_nodes_document_ptr_h2 disc_nodes_old_document_h2 disc_nodes_old_document_h3
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq2_h2
document_ptr_kinds_eq2_h3 old_document_in_heap
apply(auto)[1]
apply(cases "xb = old_document")
proof -
assume a1: "xb = old_document"
assume a2: "h2 ⊢ get_disconnected_nodes old_document →⇩r disc_nodes_old_document_h2"
assume a3: "h3 ⊢ get_disconnected_nodes old_document →⇩r remove1 child disc_nodes_old_document_h2"
assume a4: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
assume "document_ptr_kinds h2 = document_ptr_kinds h'"
assume a5: "(⋃x∈fset (object_ptr_kinds h'). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h'). set |h2 ⊢ get_disconnected_nodes x|⇩r) = {}"
have f6: "old_document |∈| document_ptr_kinds h'"
using a1 ‹xb |∈| document_ptr_kinds h'› by blast
have f7: "|h2 ⊢ get_disconnected_nodes old_document|⇩r = disc_nodes_old_document_h2"
using a2 by simp
have "x ∈ set disc_nodes_old_document_h2"
using f6 a3 a1 by (metis (no_types) ‹type_wf h'› ‹x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r›
disconnected_nodes_eq_h3 docs_neq get_disconnected_nodes_ok returns_result_eq
returns_result_select_result set_remove1_subset subsetCE)
then have "set |h' ⊢ get_child_nodes xa|⇩r ∩ set |h2 ⊢ get_disconnected_nodes xb|⇩r = {}"
using f7 f6 a5 a4 ‹xa |∈| object_ptr_kinds h'›
by fastforce
then show ?thesis
using ‹x ∈ set disc_nodes_old_document_h2› a1 a4 f7 by blast
next
assume a1: "xb ≠ old_document"
assume a2: "h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_document_ptr_h3"
assume a3: "h2 ⊢ get_disconnected_nodes old_document →⇩r disc_nodes_old_document_h2"
assume a4: "xa |∈| object_ptr_kinds h'"
assume a5: "h' ⊢ get_disconnected_nodes document_ptr →⇩r child # disc_nodes_document_ptr_h3"
assume a6: "old_document |∈| document_ptr_kinds h'"
assume a7: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
assume a8: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
assume a9: "document_ptr_kinds h2 = document_ptr_kinds h'"
assume a10: "⋀doc_ptr. old_document ≠ doc_ptr
⟹ |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
assume a11: "⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
assume a12: "(⋃x∈fset (object_ptr_kinds h'). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h'). set |h2 ⊢ get_disconnected_nodes x|⇩r) = {}"
have f13: "⋀d. d ∉ set |h' ⊢ document_ptr_kinds_M|⇩r ∨ h2 ⊢ ok get_disconnected_nodes d"
using a9 ‹type_wf h2› get_disconnected_nodes_ok
by simp
then have f14: "|h2 ⊢ get_disconnected_nodes old_document|⇩r = disc_nodes_old_document_h2"
using a6 a3 by simp
have "x ∉ set |h2 ⊢ get_disconnected_nodes xb|⇩r"
using a12 a8 a4 ‹xb |∈| document_ptr_kinds h'›
by (meson UN_I disjoint_iff_not_equal fmember.rep_eq)
then have "x = child"
using f13 a11 a10 a7 a5 a2 a1
by (metis (no_types, lifting) select_result_I2 set_ConsD)
then have "child ∉ set disc_nodes_old_document_h2"
using f14 a12 a8 a6 a4
by (metis ‹type_wf h'› adopt_node_removes_child assms(1) assms(2) type_wf
get_child_nodes_ok known_ptrs local.known_ptrs_known_ptr object_ptr_kinds_h2_eq3
object_ptr_kinds_h3_eq3 object_ptr_kinds_h_eq3 returns_result_select_result)
then show ?thesis
using ‹child ∈ set disc_nodes_old_document_h2› by fastforce
qed
qed
ultimately show ?thesis
using ‹type_wf h'› ‹known_ptrs h'› ‹a_owner_document_valid h'› heap_is_wellformed_def by blast
qed
then show "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
by auto
qed
lemma adopt_node_node_in_disconnected_nodes:
assumes wellformed: "heap_is_wellformed h"
and adopt_node: "h ⊢ adopt_node owner_document node_ptr →⇩h h'"
and "h' ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "node_ptr ∈ set disc_nodes"
proof -
obtain old_document parent_opt h2 where
old_document: "h ⊢ get_owner_document (cast node_ptr) →⇩r old_document" and
parent_opt: "h ⊢ get_parent node_ptr →⇩r parent_opt" and
h2: "h ⊢ (case parent_opt of Some parent ⇒ remove_child parent node_ptr | None ⇒ return ()) →⇩h h2"
and
h': "h2 ⊢ (if owner_document ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 node_ptr old_disc_nodes);
disc_nodes ← get_disconnected_nodes owner_document;
set_disconnected_nodes owner_document (node_ptr # disc_nodes)
} else do {
return ()
}) →⇩h h'"
using assms(2)
by(auto simp add: adopt_node_def elim!: bind_returns_heap_E
dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
pure_returns_heap_eq[rotated, OF get_parent_pure])
show ?thesis
proof (cases "owner_document = old_document")
case True
then show ?thesis
proof (insert parent_opt h2, induct parent_opt)
case None
then have "h = h'"
using h2 h' by(auto)
then show ?case
using in_disconnected_nodes_no_parent assms None old_document by blast
next
case (Some parent)
then show ?case
using remove_child_in_disconnected_nodes known_ptrs True h' assms(3) old_document by auto
qed
next
case False
then show ?thesis
using assms(3) h' list.set_intros(1) select_result_I2 set_disconnected_nodes_get_disconnected_nodes
apply(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1]
proof -
fix x and h'a and xb
assume a1: "h' ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes"
assume a2: "⋀h document_ptr disc_nodes h'. h ⊢ set_disconnected_nodes document_ptr disc_nodes →⇩h h'
⟹ h' ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
assume "h'a ⊢ set_disconnected_nodes owner_document (node_ptr # xb) →⇩h h'"
then have "node_ptr # xb = disc_nodes"
using a2 a1 by (meson returns_result_eq)
then show ?thesis
by (meson list.set_intros(1))
qed
qed
qed
end
interpretation i_adopt_node_wf?: l_adopt_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_owner_document get_parent get_parent_locs
remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr
type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs
remove heap_is_wellformed parent_child_rel
by(simp add: l_adopt_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_adopt_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
interpretation i_adopt_node_wf2?: l_adopt_node_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_owner_document get_parent get_parent_locs
remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr
type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs
remove heap_is_wellformed parent_child_rel get_root_node get_root_node_locs
by(simp add: l_adopt_node_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_adopt_node_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms[instances]
locale l_adopt_node_wf = l_heap_is_wellformed + l_known_ptrs + l_type_wf + l_adopt_node_defs
+ l_get_child_nodes_defs + l_get_disconnected_nodes_defs +
assumes adopt_node_preserves_wellformedness:
"heap_is_wellformed h ⟹ h ⊢ adopt_node document_ptr child →⇩h h' ⟹ known_ptrs h
⟹ type_wf h ⟹ heap_is_wellformed h'"
assumes adopt_node_removes_child:
"heap_is_wellformed h ⟹ h ⊢ adopt_node owner_document node_ptr →⇩h h2
⟹ h2 ⊢ get_child_nodes ptr →⇩r children ⟹ known_ptrs h
⟹ type_wf h ⟹ node_ptr ∉ set children"
assumes adopt_node_node_in_disconnected_nodes:
"heap_is_wellformed h ⟹ h ⊢ adopt_node owner_document node_ptr →⇩h h'
⟹ h' ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes
⟹ known_ptrs h ⟹ type_wf h ⟹ node_ptr ∈ set disc_nodes"
assumes adopt_node_removes_first_child: "heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ adopt_node owner_document node →⇩h h'
⟹ h ⊢ get_child_nodes ptr' →⇩r node # children
⟹ h' ⊢ get_child_nodes ptr' →⇩r children"
assumes adopt_node_document_in_heap: "heap_is_wellformed h ⟹ known_ptrs h ⟹ type_wf h
⟹ h ⊢ ok (adopt_node owner_document node)
⟹ owner_document |∈| document_ptr_kinds h"
assumes adopt_node_preserves_type_wf:
"heap_is_wellformed h ⟹ h ⊢ adopt_node document_ptr child →⇩h h' ⟹ known_ptrs h
⟹ type_wf h ⟹ type_wf h'"
assumes adopt_node_preserves_known_ptrs:
"heap_is_wellformed h ⟹ h ⊢ adopt_node document_ptr child →⇩h h' ⟹ known_ptrs h
⟹ type_wf h ⟹ known_ptrs h'"
lemma adopt_node_wf_is_l_adopt_node_wf [instances]:
"l_adopt_node_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes
get_disconnected_nodes known_ptrs adopt_node"
using heap_is_wellformed_is_l_heap_is_wellformed known_ptrs_is_l_known_ptrs
apply(auto simp add: l_adopt_node_wf_def l_adopt_node_wf_axioms_def)[1]
using adopt_node_preserves_wellformedness apply blast
using adopt_node_removes_child apply blast
using adopt_node_node_in_disconnected_nodes apply blast
using adopt_node_removes_first_child apply blast
using adopt_node_document_in_heap apply blast
using adopt_node_preserves_wellformedness apply blast
using adopt_node_preserves_wellformedness apply blast
done
subsection ‹insert\_before›
locale l_insert_before_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_adopt_node_wf +
l_set_disconnected_nodes_get_child_nodes +
l_heap_is_wellformed
begin
lemma insert_before_removes_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "ptr ≠ ptr'"
assumes "h ⊢ insert_before ptr node child →⇩h h'"
assumes "h ⊢ get_child_nodes ptr' →⇩r node # children"
shows "h' ⊢ get_child_nodes ptr' →⇩r children"
proof -
obtain owner_document h2 h3 disc_nodes reference_child where
"h ⊢ (if Some node = child then a_next_sibling node else return child) →⇩r reference_child" and
"h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document node →⇩h h2" and
"h2 ⊢ get_disconnected_nodes owner_document →⇩r disc_nodes" and
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 node disc_nodes) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr node reference_child →⇩h h'"
using assms(5)
by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
split: if_splits option.splits)
have "h2 ⊢ get_child_nodes ptr' →⇩r children"
using h2 adopt_node_removes_first_child assms(1) assms(2) assms(3) assms(6)
by simp
then have "h3 ⊢ get_child_nodes ptr' →⇩r children"
using h3
by(auto simp add: set_disconnected_nodes_get_child_nodes
dest!: reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes])
then show ?thesis
using h' assms(4)
apply(auto simp add: a_insert_node_def
elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated])[1]
by(auto simp add: set_child_nodes_get_child_nodes_different_pointers
elim!: reads_writes_separate_forwards[OF get_child_nodes_reads set_child_nodes_writes])
qed
end
locale l_insert_before_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs
+ l_insert_before_defs + l_get_child_nodes_defs +
assumes insert_before_removes_child:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ ptr ≠ ptr'
⟹ h ⊢ insert_before ptr node child →⇩h h'
⟹ h ⊢ get_child_nodes ptr' →⇩r node # children
⟹ h' ⊢ get_child_nodes ptr' →⇩r children"
interpretation i_insert_before_wf?: l_insert_before_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_parent get_parent_locs
get_child_nodes get_child_nodes_locs set_child_nodes
set_child_nodes_locs get_ancestors get_ancestors_locs
adopt_node adopt_node_locs set_disconnected_nodes
set_disconnected_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs get_owner_document insert_before
insert_before_locs append_child type_wf known_ptr known_ptrs
heap_is_wellformed parent_child_rel
by(simp add: l_insert_before_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_insert_before_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
lemma insert_before_wf_is_l_insert_before_wf [instances]:
"l_insert_before_wf heap_is_wellformed type_wf known_ptr known_ptrs insert_before get_child_nodes"
apply(auto simp add: l_insert_before_wf_def l_insert_before_wf_axioms_def instances)[1]
using insert_before_removes_child apply fast
done
locale l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_insert_before_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_set_child_nodes_get_disconnected_nodes +
l_remove_child +
l_get_root_node_wf +
l_set_disconnected_nodes_get_disconnected_nodes_wf +
l_set_disconnected_nodes_get_ancestors +
l_get_ancestors_wf +
l_get_owner_document +
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_owner_document_wf
begin
lemma insert_before_preserves_acyclitity:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ insert_before ptr node child →⇩h h'"
shows "acyclic (parent_child_rel h')"
proof -
obtain ancestors reference_child owner_document h2 h3
disconnected_nodes_h2
where
ancestors: "h ⊢ get_ancestors ptr →⇩r ancestors" and
node_not_in_ancestors: "cast node ∉ set ancestors" and
reference_child:
"h ⊢ (if Some node = child then a_next_sibling node
else return child) →⇩r reference_child" and
owner_document: "h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document node →⇩h h2" and
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document
→⇩r disconnected_nodes_h2" and
h3: "h2 ⊢ set_disconnected_nodes owner_document
(remove1 node disconnected_nodes_h2) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr node reference_child →⇩h h'"
using assms(4)
by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
split: if_splits option.splits)
have "known_ptr ptr"
by (meson get_owner_document_ptr_in_heap is_OK_returns_result_I assms
l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms owner_document)
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF adopt_node_writes h2]
using assms adopt_node_types_preserved
by(auto simp add: a_remove_child_locs_def reflp_def transp_def)
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF insert_node_writes h']
using set_child_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have object_ptr_kinds_M_eq3_h: "object_ptr_kinds h = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF adopt_node_writes h2])
using adopt_node_pointers_preserved
apply blast
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h: "⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h2 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs )
then have object_ptr_kinds_M_eq2_h: "|h ⊢ object_ptr_kinds_M|⇩r = |h2 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h: "|h ⊢ node_ptr_kinds_M|⇩r = |h2 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have "known_ptrs h2"
using assms object_ptr_kinds_M_eq3_h known_ptrs_preserved by blast
have wellformed_h2: "heap_is_wellformed h2"
using adopt_node_preserves_wellformedness[OF assms(1) h2] assms by simp
have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
unfolding a_remove_child_locs_def
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h2: "⋀ptrs. h2 ⊢ object_ptr_kinds_M →⇩r ptrs = h3 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h2: "|h2 ⊢ object_ptr_kinds_M|⇩r = |h3 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h2: "|h2 ⊢ node_ptr_kinds_M|⇩r = |h3 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h2: "|h2 ⊢ document_ptr_kinds_M|⇩r = |h3 ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h2 document_ptr_kinds_M_eq by auto
have "known_ptrs h3"
using object_ptr_kinds_M_eq3_h2 known_ptrs_preserved ‹known_ptrs h2› by blast
have object_ptr_kinds_M_eq3_h': "object_ptr_kinds h3 = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF insert_node_writes h'])
unfolding a_remove_child_locs_def
using set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h3:
"⋀ptrs. h3 ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h3:
"|h3 ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h3: "|h3 ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h3: "|h3 ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h3 document_ptr_kinds_M_eq by auto
have "known_ptrs h'"
using object_ptr_kinds_M_eq3_h' known_ptrs_preserved ‹known_ptrs h3› by blast
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. owner_document ≠ doc_ptr
⟹ h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes = h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. doc_ptr ≠ owner_document
⟹ |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disconnected_nodes_h3:
"h3 ⊢ get_disconnected_nodes owner_document →⇩r remove1 node disconnected_nodes_h2"
using h3 set_disconnected_nodes_get_disconnected_nodes
by blast
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
using set_child_nodes_get_disconnected_nodes by fast
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h2:
"⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have children_eq_h3:
"⋀ptr' children. ptr ≠ ptr'
⟹ h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
by (auto simp add: set_child_nodes_get_child_nodes_different_pointers)
then have children_eq2_h3:
"⋀ptr'. ptr ≠ ptr' ⟹ |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
obtain children_h3 where children_h3: "h3 ⊢ get_child_nodes ptr →⇩r children_h3"
using h' a_insert_node_def by auto
have children_h': "h' ⊢ get_child_nodes ptr →⇩r insert_before_list node reference_child children_h3"
using h' ‹type_wf h3› ‹known_ptr ptr›
by(auto simp add: a_insert_node_def elim!: bind_returns_heap_E2
dest!: set_child_nodes_get_child_nodes returns_result_eq[OF children_h3])
have ptr_in_heap: "ptr |∈| object_ptr_kinds h3"
using children_h3 get_child_nodes_ptr_in_heap by blast
have node_in_heap: "node |∈| node_ptr_kinds h"
using h2 adopt_node_child_in_heap by fast
have child_not_in_any_children:
"⋀p children. h2 ⊢ get_child_nodes p →⇩r children ⟹ node ∉ set children"
using assms h2 adopt_node_removes_child by auto
have "node ∈ set disconnected_nodes_h2"
using disconnected_nodes_h2 h2 adopt_node_node_in_disconnected_nodes assms(1)
‹type_wf h› ‹known_ptrs h› by blast
have node_not_in_disconnected_nodes:
"⋀d. d |∈| document_ptr_kinds h3 ⟹ node ∉ set |h3 ⊢ get_disconnected_nodes d|⇩r"
proof -
fix d
assume "d |∈| document_ptr_kinds h3"
show "node ∉ set |h3 ⊢ get_disconnected_nodes d|⇩r"
proof (cases "d = owner_document")
case True
then show ?thesis
using disconnected_nodes_h2 wellformed_h2 h3 remove_from_disconnected_nodes_removes
wellformed_h2 ‹d |∈| document_ptr_kinds h3› disconnected_nodes_h3
by fastforce
next
case False
then have
"set |h2 ⊢ get_disconnected_nodes d|⇩r ∩ set |h2 ⊢ get_disconnected_nodes owner_document|⇩r = {}"
using distinct_concat_map_E(1) wellformed_h2
by (metis (no_types, lifting) ‹d |∈| document_ptr_kinds h3› ‹type_wf h2›
disconnected_nodes_h2 document_ptr_kinds_M_def document_ptr_kinds_eq2_h2
l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok
local.heap_is_wellformed_one_disc_parent returns_result_select_result
select_result_I2)
then show ?thesis
using disconnected_nodes_eq2_h2[OF False] ‹node ∈ set disconnected_nodes_h2›
disconnected_nodes_h2 by fastforce
qed
qed
have "cast node ≠ ptr"
using ancestors node_not_in_ancestors get_ancestors_ptr
by fast
obtain ancestors_h2 where ancestors_h2: "h2 ⊢ get_ancestors ptr →⇩r ancestors_h2"
using get_ancestors_ok object_ptr_kinds_M_eq2_h2 ‹known_ptrs h2› ‹type_wf h2›
by (metis is_OK_returns_result_E object_ptr_kinds_M_eq3_h2 ptr_in_heap wellformed_h2)
have ancestors_h3: "h3 ⊢ get_ancestors ptr →⇩r ancestors_h2"
using get_ancestors_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_separate_forwards)
using ‹heap_is_wellformed h2› ancestors_h2
by (auto simp add: set_disconnected_nodes_get_ancestors)
have node_not_in_ancestors_h2: "cast node ∉ set ancestors_h2"
apply(rule get_ancestors_remains_not_in_ancestors[OF assms(1) wellformed_h2 ancestors ancestors_h2])
using adopt_node_children_subset using h2 ‹known_ptrs h› ‹ type_wf h› apply(blast)
using node_not_in_ancestors apply(blast)
using object_ptr_kinds_M_eq3_h apply(blast)
using ‹known_ptrs h› apply(blast)
using ‹type_wf h› apply(blast)
using ‹type_wf h2› by blast
have "acyclic (parent_child_rel h2)"
using wellformed_h2 by (simp add: heap_is_wellformed_def acyclic_heap_def)
then have "acyclic (parent_child_rel h3)"
by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
moreover
have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h2)⇧*}"
using adopt_node_removes_child
using ancestors node_not_in_ancestors
using ‹known_ptrs h2› ‹type_wf h2› ancestors_h2 local.get_ancestors_parent_child_rel
node_not_in_ancestors_h2 wellformed_h2
by blast
then have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h3)⇧*}"
by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
moreover have "parent_child_rel h'
= insert (ptr, cast node) ((parent_child_rel h3))"
using children_h3 children_h' ptr_in_heap
apply(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h' children_eq2_h3
insert_before_list_node_in_set)[1]
apply (metis (no_types, lifting) children_eq2_h3 insert_before_list_in_set select_result_I2)
by (metis (no_types, lifting) children_eq2_h3 imageI insert_before_list_in_set select_result_I2)
ultimately show "acyclic (parent_child_rel h')"
by (auto simp add: heap_is_wellformed_def)
qed
lemma insert_before_heap_is_wellformed_preserved:
assumes wellformed: "heap_is_wellformed h"
and insert_before: "h ⊢ insert_before ptr node child →⇩h h'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs h'"
proof -
obtain ancestors reference_child owner_document h2 h3 disconnected_nodes_h2 where
ancestors: "h ⊢ get_ancestors ptr →⇩r ancestors" and
node_not_in_ancestors: "cast node ∉ set ancestors" and
reference_child:
"h ⊢ (if Some node = child then a_next_sibling node else return child) →⇩r reference_child" and
owner_document: "h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document node →⇩h h2" and
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h2" and
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr node reference_child →⇩h h'"
using assms(2)
by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
split: if_splits option.splits)
have "known_ptr ptr"
by (meson get_owner_document_ptr_in_heap is_OK_returns_result_I known_ptrs
l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms owner_document)
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF adopt_node_writes h2]
using type_wf adopt_node_types_preserved
by(auto simp add: a_remove_child_locs_def reflp_def transp_def)
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
then show "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF insert_node_writes h']
using set_child_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have object_ptr_kinds_M_eq3_h: "object_ptr_kinds h = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF adopt_node_writes h2])
using adopt_node_pointers_preserved
apply blast
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h: "⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h2 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs )
then have object_ptr_kinds_M_eq2_h: "|h ⊢ object_ptr_kinds_M|⇩r = |h2 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h: "|h ⊢ node_ptr_kinds_M|⇩r = |h2 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have "known_ptrs h2"
using known_ptrs object_ptr_kinds_M_eq3_h known_ptrs_preserved by blast
have wellformed_h2: "heap_is_wellformed h2"
using adopt_node_preserves_wellformedness[OF wellformed h2] known_ptrs type_wf .
have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
unfolding a_remove_child_locs_def
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h2: "⋀ptrs. h2 ⊢ object_ptr_kinds_M →⇩r ptrs = h3 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h2: "|h2 ⊢ object_ptr_kinds_M|⇩r = |h3 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h2: "|h2 ⊢ node_ptr_kinds_M|⇩r = |h3 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h2: "|h2 ⊢ document_ptr_kinds_M|⇩r = |h3 ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h2 document_ptr_kinds_M_eq by auto
have "known_ptrs h3"
using object_ptr_kinds_M_eq3_h2 known_ptrs_preserved ‹known_ptrs h2› by blast
have object_ptr_kinds_M_eq3_h': "object_ptr_kinds h3 = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF insert_node_writes h'])
unfolding a_remove_child_locs_def
using set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h3:
"⋀ptrs. h3 ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h3:
"|h3 ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h3: "|h3 ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h3: "|h3 ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h3 document_ptr_kinds_M_eq by auto
show "known_ptrs h'"
using object_ptr_kinds_M_eq3_h' known_ptrs_preserved ‹known_ptrs h3› by blast
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. owner_document ≠ doc_ptr
⟹ h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes = h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. doc_ptr ≠ owner_document
⟹ |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disconnected_nodes_h3:
"h3 ⊢ get_disconnected_nodes owner_document →⇩r remove1 node disconnected_nodes_h2"
using h3 set_disconnected_nodes_get_disconnected_nodes
by blast
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
using set_child_nodes_get_disconnected_nodes by fast
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h2:
"⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have children_eq_h3:
"⋀ptr' children. ptr ≠ ptr'
⟹ h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
by (auto simp add: set_child_nodes_get_child_nodes_different_pointers)
then have children_eq2_h3:
"⋀ptr'. ptr ≠ ptr' ⟹ |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
obtain children_h3 where children_h3: "h3 ⊢ get_child_nodes ptr →⇩r children_h3"
using h' a_insert_node_def by auto
have children_h': "h' ⊢ get_child_nodes ptr →⇩r insert_before_list node reference_child children_h3"
using h' ‹type_wf h3› ‹known_ptr ptr›
by(auto simp add: a_insert_node_def elim!: bind_returns_heap_E2
dest!: set_child_nodes_get_child_nodes returns_result_eq[OF children_h3])
have ptr_in_heap: "ptr |∈| object_ptr_kinds h3"
using children_h3 get_child_nodes_ptr_in_heap by blast
have node_in_heap: "node |∈| node_ptr_kinds h"
using h2 adopt_node_child_in_heap by fast
have child_not_in_any_children:
"⋀p children. h2 ⊢ get_child_nodes p →⇩r children ⟹ node ∉ set children"
using wellformed h2 adopt_node_removes_child ‹type_wf h› ‹known_ptrs h› by auto
have "node ∈ set disconnected_nodes_h2"
using disconnected_nodes_h2 h2 adopt_node_node_in_disconnected_nodes assms(1)
‹type_wf h› ‹known_ptrs h› by blast
have node_not_in_disconnected_nodes:
"⋀d. d |∈| document_ptr_kinds h3 ⟹ node ∉ set |h3 ⊢ get_disconnected_nodes d|⇩r"
proof -
fix d
assume "d |∈| document_ptr_kinds h3"
show "node ∉ set |h3 ⊢ get_disconnected_nodes d|⇩r"
proof (cases "d = owner_document")
case True
then show ?thesis
using disconnected_nodes_h2 wellformed_h2 h3 remove_from_disconnected_nodes_removes
wellformed_h2 ‹d |∈| document_ptr_kinds h3› disconnected_nodes_h3
by fastforce
next
case False
then have
"set |h2 ⊢ get_disconnected_nodes d|⇩r ∩ set |h2 ⊢ get_disconnected_nodes owner_document|⇩r = {}"
using distinct_concat_map_E(1) wellformed_h2
by (metis (no_types, lifting) ‹d |∈| document_ptr_kinds h3› ‹type_wf h2›
disconnected_nodes_h2 document_ptr_kinds_M_def document_ptr_kinds_eq2_h2
l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok
local.heap_is_wellformed_one_disc_parent returns_result_select_result
select_result_I2)
then show ?thesis
using disconnected_nodes_eq2_h2[OF False] ‹node ∈ set disconnected_nodes_h2›
disconnected_nodes_h2 by fastforce
qed
qed
have "cast node ≠ ptr"
using ancestors node_not_in_ancestors get_ancestors_ptr
by fast
obtain ancestors_h2 where ancestors_h2: "h2 ⊢ get_ancestors ptr →⇩r ancestors_h2"
using get_ancestors_ok object_ptr_kinds_M_eq2_h2 ‹known_ptrs h2› ‹type_wf h2›
by (metis is_OK_returns_result_E object_ptr_kinds_M_eq3_h2 ptr_in_heap wellformed_h2)
have ancestors_h3: "h3 ⊢ get_ancestors ptr →⇩r ancestors_h2"
using get_ancestors_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_separate_forwards)
using ‹heap_is_wellformed h2› ancestors_h2
by (auto simp add: set_disconnected_nodes_get_ancestors)
have node_not_in_ancestors_h2: "cast node ∉ set ancestors_h2"
apply(rule get_ancestors_remains_not_in_ancestors[OF assms(1) wellformed_h2 ancestors ancestors_h2])
using adopt_node_children_subset using h2 ‹known_ptrs h› ‹ type_wf h› apply(blast)
using node_not_in_ancestors apply(blast)
using object_ptr_kinds_M_eq3_h apply(blast)
using ‹known_ptrs h› apply(blast)
using ‹type_wf h› apply(blast)
using ‹type_wf h2› by blast
moreover have "a_acyclic_heap h'"
proof -
have "acyclic (parent_child_rel h2)"
using wellformed_h2 by (simp add: heap_is_wellformed_def acyclic_heap_def)
then have "acyclic (parent_child_rel h3)"
by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
moreover have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h2)⇧*}"
using get_ancestors_parent_child_rel node_not_in_ancestors_h2 ‹known_ptrs h2› ‹type_wf h2›
using ancestors_h2 wellformed_h2 by blast
then have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h3)⇧*}"
by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
moreover have "parent_child_rel h' = insert (ptr, cast node) ((parent_child_rel h3))"
using children_h3 children_h' ptr_in_heap
apply(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h' children_eq2_h3
insert_before_list_node_in_set)[1]
apply (metis (no_types, lifting) children_eq2_h3 insert_before_list_in_set select_result_I2)
by (metis (no_types, lifting) children_eq2_h3 imageI insert_before_list_in_set select_result_I2)
ultimately show ?thesis
by(auto simp add: acyclic_heap_def)
qed
moreover have "a_all_ptrs_in_heap h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
have "a_all_ptrs_in_heap h'"
proof -
have "a_all_ptrs_in_heap h3"
using ‹a_all_ptrs_in_heap h2›
apply(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_M_eq2_h2 node_ptr_kinds_eq2_h2
children_eq_h2)[1]
using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3
using node_ptr_kinds_eq2_h2 apply auto[1]
apply (metis ‹known_ptrs h2› ‹type_wf h3› children_eq_h2 local.get_child_nodes_ok
local.heap_is_wellformed_children_in_heap local.known_ptrs_known_ptr object_ptr_kinds_M_eq3_h2
returns_result_select_result wellformed_h2)
by (metis (no_types, lifting) disconnected_nodes_eq2_h2 disconnected_nodes_h2
disconnected_nodes_h3 document_ptr_kinds_commutes finite_set_in node_ptr_kinds_commutes
object_ptr_kinds_M_eq3_h2 select_result_I2 set_remove1_subset subsetD)
have "set children_h3 ⊆ set |h' ⊢ node_ptr_kinds_M|⇩r"
using children_h3 ‹a_all_ptrs_in_heap h3›
apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq2_h3)[1]
by (metis children_eq_h2 l_heap_is_wellformed.heap_is_wellformed_children_in_heap
local.l_heap_is_wellformed_axioms node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h'
object_ptr_kinds_M_eq3_h2 wellformed_h2)
then have "set (insert_before_list node reference_child children_h3) ⊆ set |h' ⊢ node_ptr_kinds_M|⇩r"
using node_in_heap
apply(auto simp add: node_ptr_kinds_eq2_h node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3)[1]
by (metis (no_types, hide_lams) contra_subsetD finite_set_in insert_before_list_in_set
node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h'
object_ptr_kinds_M_eq3_h2)
then show ?thesis
using ‹a_all_ptrs_in_heap h3›
apply(auto simp add: object_ptr_kinds_M_eq3_h' a_all_ptrs_in_heap_def node_ptr_kinds_def
node_ptr_kinds_eq2_h3 disconnected_nodes_eq_h3)[1]
using children_eq_h3 children_h'
apply (metis (no_types, lifting) children_eq2_h3 finite_set_in select_result_I2 subsetD)
by (metis (no_types) ‹type_wf h'› disconnected_nodes_eq2_h3 disconnected_nodes_eq_h3
finite_set_in is_OK_returns_result_I local.get_disconnected_nodes_ok
local.get_disconnected_nodes_ptr_in_heap returns_result_select_result subsetD)
qed
moreover have "a_distinct_lists h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h3"
proof(auto simp add: a_distinct_lists_def object_ptr_kinds_M_eq2_h2 document_ptr_kinds_eq2_h2
children_eq2_h2 intro!: distinct_concat_map_I)[1]
fix x
assume 1: "x |∈| document_ptr_kinds h3"
and 2: "distinct (concat (map (λdocument_ptr. |h2 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
show "distinct |h3 ⊢ get_disconnected_nodes x|⇩r"
using distinct_concat_map_E(2)[OF 2] select_result_I2[OF disconnected_nodes_h3]
disconnected_nodes_eq2_h2 select_result_I2[OF disconnected_nodes_h2] 1
by (metis (full_types) distinct_remove1 finite_fset fmember.rep_eq set_sorted_list_of_set)
next
fix x y xa
assume 1: "distinct (concat (map (λdocument_ptr. |h2 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and 2: "x |∈| document_ptr_kinds h3"
and 3: "y |∈| document_ptr_kinds h3"
and 4: "x ≠ y"
and 5: "xa ∈ set |h3 ⊢ get_disconnected_nodes x|⇩r"
and 6: "xa ∈ set |h3 ⊢ get_disconnected_nodes y|⇩r"
show False
proof (cases "x = owner_document")
case True
then have "y ≠ owner_document"
using 4 by simp
show ?thesis
using distinct_concat_map_E(1)[OF 1]
using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3] select_result_I2[OF disconnected_nodes_h2]
apply(auto simp add: True disconnected_nodes_eq2_h2[OF ‹y ≠ owner_document›])[1]
by (metis (no_types, hide_lams) disconnected_nodes_eq2_h2 disjoint_iff_not_equal notin_set_remove1)
next
case False
then show ?thesis
proof (cases "y = owner_document")
case True
then show ?thesis
using distinct_concat_map_E(1)[OF 1]
using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3] select_result_I2[OF disconnected_nodes_h2]
apply(auto simp add: True disconnected_nodes_eq2_h2[OF ‹x ≠ owner_document›])[1]
by (metis (no_types, hide_lams) disconnected_nodes_eq2_h2 disjoint_iff_not_equal notin_set_remove1)
next
case False
then show ?thesis
using distinct_concat_map_E(1)[OF 1, simplified, OF 2 3 4] 5 6
using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3
disjoint_iff_not_equal finite_fset fmember.rep_eq notin_set_remove1 select_result_I2
set_sorted_list_of_set
by (metis (no_types, lifting))
qed
qed
next
fix x xa xb
assume 1: "(⋃x∈fset (object_ptr_kinds h3). set |h3 ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h3). set |h2 ⊢ get_disconnected_nodes x|⇩r) = {}"
and 2: "xa |∈| object_ptr_kinds h3"
and 3: "x ∈ set |h3 ⊢ get_child_nodes xa|⇩r"
and 4: "xb |∈| document_ptr_kinds h3"
and 5: "x ∈ set |h3 ⊢ get_disconnected_nodes xb|⇩r"
have 6: "set |h3 ⊢ get_child_nodes xa|⇩r ∩ set |h2 ⊢ get_disconnected_nodes xb|⇩r = {}"
using 1 2 4
by (metis ‹type_wf h2› children_eq2_h2 document_ptr_kinds_commutes known_ptrs
local.get_child_nodes_ok local.get_disconnected_nodes_ok
local.heap_is_wellformed_children_disc_nodes_different local.known_ptrs_known_ptr
object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h2 returns_result_select_result
wellformed_h2)
show False
proof (cases "xb = owner_document")
case True
then show ?thesis
using select_result_I2[OF disconnected_nodes_h3,folded select_result_I2[OF disconnected_nodes_h2]]
by (metis (no_types, lifting) "3" "5" "6" disjoint_iff_not_equal notin_set_remove1)
next
case False
show ?thesis
using 2 3 4 5 6 unfolding disconnected_nodes_eq2_h2[OF False] by auto
qed
qed
then have "a_distinct_lists h'"
proof(auto simp add: a_distinct_lists_def document_ptr_kinds_eq2_h3 object_ptr_kinds_M_eq2_h3
disconnected_nodes_eq2_h3 intro!: distinct_concat_map_I)[1]
fix x
assume 1: "distinct (concat (map (λptr. |h3 ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))" and
2: "x |∈| object_ptr_kinds h'"
have 3: "⋀p. p |∈| object_ptr_kinds h' ⟹ distinct |h3 ⊢ get_child_nodes p|⇩r"
using 1 by (auto elim: distinct_concat_map_E)
show "distinct |h' ⊢ get_child_nodes x|⇩r"
proof(cases "ptr = x")
case True
show ?thesis
using 3[OF 2] children_h3 children_h'
by(auto simp add: True insert_before_list_distinct
dest: child_not_in_any_children[unfolded children_eq_h2])
next
case False
show ?thesis
using children_eq2_h3[OF False] 3[OF 2] by auto
qed
next
fix x y xa
assume 1: "distinct (concat (map (λptr. |h3 ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))"
and 2: "x |∈| object_ptr_kinds h'"
and 3: "y |∈| object_ptr_kinds h'"
and 4: "x ≠ y"
and 5: "xa ∈ set |h' ⊢ get_child_nodes x|⇩r"
and 6: "xa ∈ set |h' ⊢ get_child_nodes y|⇩r"
have 7:"set |h3 ⊢ get_child_nodes x|⇩r ∩ set |h3 ⊢ get_child_nodes y|⇩r = {}"
using distinct_concat_map_E(1)[OF 1] 2 3 4 by auto
show False
proof (cases "ptr = x")
case True
then have "ptr ≠ y"
using 4 by simp
then show ?thesis
using children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6
apply(auto simp add: True children_eq2_h3[OF ‹ptr ≠ y›])[1]
by (metis (no_types, hide_lams) "3" "7" ‹type_wf h3› children_eq2_h3 disjoint_iff_not_equal
get_child_nodes_ok insert_before_list_in_set known_ptrs local.known_ptrs_known_ptr
object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h'
object_ptr_kinds_M_eq3_h2 returns_result_select_result select_result_I2)
next
case False
then show ?thesis
proof (cases "ptr = y")
case True
then show ?thesis
using children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6
apply(auto simp add: True children_eq2_h3[OF ‹ptr ≠ x›])[1]
by (metis (no_types, hide_lams) "2" "4" "7" IntI ‹known_ptrs h3› ‹type_wf h'›
children_eq_h3 empty_iff insert_before_list_in_set local.get_child_nodes_ok
local.known_ptrs_known_ptr object_ptr_kinds_M_eq3_h'
returns_result_select_result select_result_I2)
next
case False
then show ?thesis
using children_eq2_h3[OF ‹ptr ≠ x›] children_eq2_h3[OF ‹ptr ≠ y›] 5 6 7 by auto
qed
qed
next
fix x xa xb
assume 1: " (⋃x∈fset (object_ptr_kinds h'). set |h3 ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h'). set |h' ⊢ get_disconnected_nodes x|⇩r) = {} "
and 2: "xa |∈| object_ptr_kinds h'"
and 3: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 4: "xb |∈| document_ptr_kinds h'"
and 5: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
have 6: "set |h3 ⊢ get_child_nodes xa|⇩r ∩ set |h' ⊢ get_disconnected_nodes xb|⇩r = {}"
using 1 2 3 4 5
proof -
have "∀h d. ¬ type_wf h ∨ d |∉| document_ptr_kinds h ∨ h ⊢ ok get_disconnected_nodes d"
using local.get_disconnected_nodes_ok by satx
then have "h' ⊢ ok get_disconnected_nodes xb"
using "4" ‹type_wf h'› by fastforce
then have f1: "h3 ⊢ get_disconnected_nodes xb →⇩r |h' ⊢ get_disconnected_nodes xb|⇩r"
by (simp add: disconnected_nodes_eq_h3)
have "xa |∈| object_ptr_kinds h3"
using "2" object_ptr_kinds_M_eq3_h' by blast
then show ?thesis
using f1 ‹local.a_distinct_lists h3› local.distinct_lists_no_parent by fastforce
qed
show False
proof (cases "ptr = xa")
case True
show ?thesis
using 6 node_not_in_disconnected_nodes 3 4 5 select_result_I2[OF children_h']
select_result_I2[OF children_h3] True disconnected_nodes_eq2_h3
by (metis (no_types, lifting) "2" DocumentMonad.ptr_kinds_ptr_kinds_M
‹a_distinct_lists h3› ‹type_wf h'› disconnected_nodes_eq_h3
distinct_lists_no_parent document_ptr_kinds_eq2_h3 get_disconnected_nodes_ok
insert_before_list_in_set object_ptr_kinds_M_eq3_h' returns_result_select_result)
next
case False
then show ?thesis
using 1 2 3 4 5 children_eq2_h3[OF False] by fastforce
qed
qed
moreover have "a_owner_document_valid h2"
using wellformed_h2 by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
apply(auto simp add: a_owner_document_valid_def object_ptr_kinds_M_eq2_h2
object_ptr_kinds_M_eq2_h3 node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3
document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3 children_eq2_h2)[1]
apply(auto simp add: document_ptr_kinds_eq2_h2[simplified] document_ptr_kinds_eq2_h3[simplified]
object_ptr_kinds_M_eq2_h2[simplified] object_ptr_kinds_M_eq2_h3[simplified]
node_ptr_kinds_eq2_h2[simplified] node_ptr_kinds_eq2_h3[simplified])[1]
apply(auto simp add: disconnected_nodes_eq2_h3[symmetric])[1]
by (smt children_eq2_h3 children_h' children_h3 disconnected_nodes_eq2_h2 disconnected_nodes_h2
disconnected_nodes_h3 finite_set_in in_set_remove1 insert_before_list_in_set object_ptr_kinds_M_eq3_h'
ptr_in_heap select_result_I2)
ultimately show "heap_is_wellformed h'"
by (simp add: heap_is_wellformed_def)
qed
lemma adopt_node_children_remain_distinct:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ adopt_node owner_document node_ptr →⇩h h'"
shows "⋀ptr' children'.
h' ⊢ get_child_nodes ptr' →⇩r children' ⟹ distinct children'"
using assms(1) assms(2) assms(3) assms(4) local.adopt_node_preserves_wellformedness
local.heap_is_wellformed_children_distinct
by blast
lemma insert_node_children_remain_distinct:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ a_insert_node ptr new_child reference_child_opt →⇩h h'"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "new_child ∉ set children"
shows "⋀children'.
h' ⊢ get_child_nodes ptr →⇩r children' ⟹ distinct children'"
proof -
fix children'
assume a1: "h' ⊢ get_child_nodes ptr →⇩r children'"
have "h' ⊢ get_child_nodes ptr →⇩r (insert_before_list new_child reference_child_opt children)"
using assms(4) assms(5) apply(auto simp add: a_insert_node_def elim!: bind_returns_heap_E)[1]
using returns_result_eq set_child_nodes_get_child_nodes assms(2) assms(3)
by (metis is_OK_returns_result_I local.get_child_nodes_ptr_in_heap local.get_child_nodes_pure
local.known_ptrs_known_ptr pure_returns_heap_eq)
moreover have "a_distinct_lists h"
using assms local.heap_is_wellformed_def by blast
then have "⋀children. h ⊢ get_child_nodes ptr →⇩r children
⟹ distinct children"
using assms local.heap_is_wellformed_children_distinct by blast
ultimately show "h' ⊢ get_child_nodes ptr →⇩r children' ⟹ distinct children'"
using assms(5) assms(6) insert_before_list_distinct returns_result_eq by fastforce
qed
lemma insert_before_children_remain_distinct:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ insert_before ptr new_child child_opt →⇩h h'"
shows "⋀ptr' children'.
h' ⊢ get_child_nodes ptr' →⇩r children' ⟹ distinct children'"
proof -
obtain reference_child owner_document h2 h3 disconnected_nodes_h2 where
reference_child:
"h ⊢ (if Some new_child = child_opt then a_next_sibling new_child else return child_opt) →⇩r reference_child" and
owner_document: "h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document new_child →⇩h h2" and
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h2" and
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 new_child disconnected_nodes_h2) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr new_child reference_child →⇩h h'"
using assms(4)
by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
split: if_splits option.splits)
have "⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children
⟹ distinct children"
using adopt_node_children_remain_distinct
using assms(1) assms(2) assms(3) h2
by blast
moreover have "⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children
⟹ new_child ∉ set children"
using adopt_node_removes_child
using assms(1) assms(2) assms(3) h2
by blast
moreover have "⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children = h3 ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_child_nodes)
ultimately show "⋀ptr children. h' ⊢ get_child_nodes ptr →⇩r children
⟹ distinct children"
using insert_node_children_remain_distinct
by (meson assms(1) assms(2) assms(3) assms(4) insert_before_heap_is_wellformed_preserved(1)
local.heap_is_wellformed_children_distinct)
qed
lemma insert_before_removes_child:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ insert_before ptr node child →⇩h h'"
assumes "ptr ≠ ptr'"
shows "⋀children'. h' ⊢ get_child_nodes ptr' →⇩r children' ⟹ node ∉ set children'"
proof -
fix children'
assume a1: "h' ⊢ get_child_nodes ptr' →⇩r children'"
obtain ancestors reference_child owner_document h2 h3 disconnected_nodes_h2 where
ancestors: "h ⊢ get_ancestors ptr →⇩r ancestors" and
node_not_in_ancestors: "cast node ∉ set ancestors" and
reference_child:
"h ⊢ (if Some node = child then a_next_sibling node else return child) →⇩r reference_child" and
owner_document: "h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document node →⇩h h2" and
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h2" and
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr node reference_child →⇩h h'"
using assms(4)
by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
split: if_splits option.splits)
have "known_ptr ptr"
by (meson get_owner_document_ptr_in_heap is_OK_returns_result_I assms(2)
l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms owner_document)
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF adopt_node_writes h2]
using assms(3) adopt_node_types_preserved
by(auto simp add: a_remove_child_locs_def reflp_def transp_def)
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF insert_node_writes h']
using set_child_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have object_ptr_kinds_M_eq3_h: "object_ptr_kinds h = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF adopt_node_writes h2])
using adopt_node_pointers_preserved
apply blast
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h: "⋀ptrs. h ⊢ object_ptr_kinds_M →⇩r ptrs = h2 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs )
then have object_ptr_kinds_M_eq2_h: "|h ⊢ object_ptr_kinds_M|⇩r = |h2 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h: "|h ⊢ node_ptr_kinds_M|⇩r = |h2 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have "known_ptrs h2"
using assms object_ptr_kinds_M_eq3_h known_ptrs_preserved by blast
have wellformed_h2: "heap_is_wellformed h2"
using adopt_node_preserves_wellformedness[OF assms(1) h2] assms by simp
have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
unfolding a_remove_child_locs_def
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h2:
"⋀ptrs. h2 ⊢ object_ptr_kinds_M →⇩r ptrs = h3 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h2: "|h2 ⊢ object_ptr_kinds_M|⇩r = |h3 ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h2: "|h2 ⊢ node_ptr_kinds_M|⇩r = |h3 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h2: "|h2 ⊢ document_ptr_kinds_M|⇩r = |h3 ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h2 document_ptr_kinds_M_eq by auto
have "known_ptrs h3"
using object_ptr_kinds_M_eq3_h2 known_ptrs_preserved ‹known_ptrs h2› by blast
have object_ptr_kinds_M_eq3_h': "object_ptr_kinds h3 = object_ptr_kinds h'"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF insert_node_writes h'])
unfolding a_remove_child_locs_def
using set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h3:
"⋀ptrs. h3 ⊢ object_ptr_kinds_M →⇩r ptrs = h' ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_M_eq2_h3:
"|h3 ⊢ object_ptr_kinds_M|⇩r = |h' ⊢ object_ptr_kinds_M|⇩r"
by simp
then have node_ptr_kinds_eq2_h3: "|h3 ⊢ node_ptr_kinds_M|⇩r = |h' ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
have document_ptr_kinds_eq2_h3: "|h3 ⊢ document_ptr_kinds_M|⇩r = |h' ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_M_eq2_h3 document_ptr_kinds_M_eq by auto
have "known_ptrs h'"
using object_ptr_kinds_M_eq3_h' known_ptrs_preserved ‹known_ptrs h3› by blast
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. owner_document ≠ doc_ptr
⟹ h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes =
h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. doc_ptr ≠ owner_document
⟹ |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disconnected_nodes_h3:
"h3 ⊢ get_disconnected_nodes owner_document →⇩r remove1 node disconnected_nodes_h2"
using h3 set_disconnected_nodes_get_disconnected_nodes
by blast
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
using set_child_nodes_get_disconnected_nodes by fast
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h2:
"⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have children_eq_h3:
"⋀ptr' children. ptr ≠ ptr'
⟹ h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads insert_node_writes h'
apply(rule reads_writes_preserved)
by (auto simp add: set_child_nodes_get_child_nodes_different_pointers)
then have children_eq2_h3:
"⋀ptr'. ptr ≠ ptr' ⟹ |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
obtain children_h3 where children_h3: "h3 ⊢ get_child_nodes ptr →⇩r children_h3"
using h' a_insert_node_def by auto
have children_h': "h' ⊢ get_child_nodes ptr →⇩r insert_before_list node reference_child children_h3"
using h' ‹type_wf h3› ‹known_ptr ptr›
by(auto simp add: a_insert_node_def elim!: bind_returns_heap_E2
dest!: set_child_nodes_get_child_nodes returns_result_eq[OF children_h3])
have ptr_in_heap: "ptr |∈| object_ptr_kinds h3"
using children_h3 get_child_nodes_ptr_in_heap by blast
have node_in_heap: "node |∈| node_ptr_kinds h"
using h2 adopt_node_child_in_heap by fast
have child_not_in_any_children:
"⋀p children. h2 ⊢ get_child_nodes p →⇩r children ⟹ node ∉ set children"
using assms(1) assms(2) assms(3) h2 local.adopt_node_removes_child by blast
show "node ∉ set children'"
using a1 assms(5) child_not_in_any_children children_eq_h2 children_eq_h3 by blast
qed
lemma ensure_pre_insertion_validity_ok:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "ptr |∈| object_ptr_kinds h"
assumes "¬is_character_data_ptr_kind parent"
assumes "cast node ∉ set |h ⊢ get_ancestors parent|⇩r"
assumes "h ⊢ get_parent ref →⇩r Some parent"
assumes "is_document_ptr parent ⟹ h ⊢ get_child_nodes parent →⇩r []"
assumes "is_document_ptr parent ⟹ ¬is_character_data_ptr_kind node"
shows "h ⊢ ok (a_ensure_pre_insertion_validity node parent (Some ref))"
proof -
have "h ⊢ (if is_character_data_ptr_kind parent
then error HierarchyRequestError else return ()) →⇩r ()"
using assms
by (simp add: assms(4))
moreover have "h ⊢ do {
ancestors ← get_ancestors parent;
(if cast node ∈ set ancestors then error HierarchyRequestError else return ())
} →⇩r ()"
using assms(6)
apply(auto intro!: bind_pure_returns_result_I)[1]
using assms(1) assms(2) assms(3) assms(7) local.get_ancestors_ok local.get_parent_parent_in_heap
by auto
moreover have "h ⊢ do {
(case Some ref of
Some child ⇒ do {
child_parent ← get_parent child;
(if child_parent ≠ Some parent then error NotFoundError else return ())}
| None ⇒ return ())
} →⇩r ()"
using assms(7)
by(auto split: option.splits)
moreover have "h ⊢ do {
children ← get_child_nodes parent;
(if children ≠ [] ∧ is_document_ptr parent
then error HierarchyRequestError else return ())
} →⇩r ()"
using assms(8)
by (smt assms(5) assms(7) bind_pure_returns_result_I2 calculation(1) is_OK_returns_result_I
local.get_child_nodes_pure local.get_parent_child_dual returns_result_eq)
moreover have "h ⊢ do {
(if is_character_data_ptr node ∧ is_document_ptr parent
then error HierarchyRequestError else return ())
} →⇩r ()"
using assms
using is_character_data_ptr_kind_none by force
ultimately show ?thesis
unfolding a_ensure_pre_insertion_validity_def
apply(intro bind_is_OK_pure_I)
apply auto[1]
apply auto[1]
apply auto[1]
using assms(1) assms(2) assms(3) assms(7) local.get_ancestors_ok local.get_parent_parent_in_heap
apply blast
apply auto[1]
apply auto[1]
using assms(6)
apply auto[1]
using assms(1) assms(2) assms(3) assms(7) local.get_ancestors_ok local.get_parent_parent_in_heap
apply auto[1]
apply (smt bind_returns_heap_E is_OK_returns_heap_E local.get_parent_pure pure_def
pure_returns_heap_eq return_returns_heap returns_result_eq)
apply(blast)
using local.get_child_nodes_pure
apply blast
apply (meson assms(7) is_OK_returns_result_I local.get_parent_child_dual)
apply (simp)
apply (smt assms(5) assms(8) is_OK_returns_result_I returns_result_eq)
by(auto)
qed
end
locale l_insert_before_wf2 = l_type_wf + l_known_ptrs + l_insert_before_defs
+ l_heap_is_wellformed_defs + l_get_child_nodes_defs + l_remove_defs +
assumes insert_before_preserves_type_wf:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ insert_before ptr child ref →⇩h h'
⟹ type_wf h'"
assumes insert_before_preserves_known_ptrs:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ insert_before ptr child ref →⇩h h'
⟹ known_ptrs h'"
assumes insert_before_heap_is_wellformed_preserved:
"type_wf h ⟹ known_ptrs h ⟹ heap_is_wellformed h ⟹ h ⊢ insert_before ptr child ref →⇩h h'
⟹ heap_is_wellformed h'"
interpretation i_insert_before_wf2?: l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_parent get_parent_locs
get_child_nodes get_child_nodes_locs set_child_nodes
set_child_nodes_locs get_ancestors get_ancestors_locs
adopt_node adopt_node_locs set_disconnected_nodes
set_disconnected_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs get_owner_document insert_before
insert_before_locs append_child type_wf known_ptr known_ptrs
heap_is_wellformed parent_child_rel remove_child
remove_child_locs get_root_node get_root_node_locs
by(simp add: l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
lemma insert_before_wf2_is_l_insert_before_wf2 [instances]:
"l_insert_before_wf2 type_wf known_ptr known_ptrs insert_before heap_is_wellformed"
apply(auto simp add: l_insert_before_wf2_def l_insert_before_wf2_axioms_def instances)[1]
using insert_before_heap_is_wellformed_preserved apply(fast, fast, fast)
done
locale l_insert_before_wf3⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_set_child_nodes_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_remove_child_wf2
begin
lemma next_sibling_ok:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "node_ptr |∈| node_ptr_kinds h"
shows "h ⊢ ok (a_next_sibling node_ptr)"
proof -
have "known_ptr (cast node_ptr)"
using assms(2) assms(4) local.known_ptrs_known_ptr node_ptr_kinds_commutes by blast
then show ?thesis
using assms
apply(auto simp add: a_next_sibling_def intro!: bind_is_OK_pure_I split: option.splits list.splits)[1]
using get_child_nodes_ok local.get_parent_parent_in_heap local.known_ptrs_known_ptr by blast
qed
lemma remove_child_ok:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "child ∈ set children"
shows "h ⊢ ok (remove_child ptr child)"
proof -
have "ptr |∈| object_ptr_kinds h"
using assms(4) local.get_child_nodes_ptr_in_heap by blast
have "child |∈| node_ptr_kinds h"
using assms(1) assms(4) assms(5) local.heap_is_wellformed_children_in_heap by blast
have "¬is_character_data_ptr ptr"
proof (rule ccontr, simp)
assume "is_character_data_ptr ptr"
then have "h ⊢ get_child_nodes ptr →⇩r []"
using ‹ptr |∈| object_ptr_kinds h›
apply(simp add: get_child_nodes_def a_get_child_nodes_tups_def)
apply(split invoke_splits)+
by(auto simp add: get_child_nodes⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r_def intro!: bind_pure_returns_result_I split: option.splits)
then
show False
using assms returns_result_eq by fastforce
qed
have "is_character_data_ptr child ⟹ ¬is_document_ptr_kind ptr"
proof (rule ccontr, simp)
assume "is_character_data_ptr⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r child"
and "is_document_ptr_kind ptr"
then show False
using assms
using ‹ptr |∈| object_ptr_kinds h›
apply(simp add: get_child_nodes_def a_get_child_nodes_tups_def)
apply(split invoke_splits)+
apply(auto split: option.splits)[1]
apply (meson invoke_empty is_OK_returns_result_I)
apply (meson invoke_empty is_OK_returns_result_I)
by(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2 split: option.splits)
qed
obtain owner_document where
owner_document: "h ⊢ get_owner_document (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r owner_document"
by (meson ‹child |∈| node_ptr_kinds h› assms(1) assms(2) assms(3) is_OK_returns_result_E
local.get_owner_document_ok node_ptr_kinds_commutes)
obtain disconnected_nodes_h where
disconnected_nodes_h: "h ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h"
by (meson assms(1) assms(2) assms(3) is_OK_returns_result_E local.get_disconnected_nodes_ok
local.get_owner_document_owner_document_in_heap owner_document)
obtain h2 where
h2: "h ⊢ set_disconnected_nodes owner_document (child # disconnected_nodes_h) →⇩h h2"
by (meson assms(1) assms(2) assms(3) is_OK_returns_heap_E
l_set_disconnected_nodes.set_disconnected_nodes_ok local.get_owner_document_owner_document_in_heap
local.l_set_disconnected_nodes_axioms owner_document)
have "known_ptr ptr"
using assms(2) assms(4) local.known_ptrs_known_ptr
using ‹ptr |∈| object_ptr_kinds h› by blast
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h2]
using set_disconnected_nodes_types_preserved assms(3)
by(auto simp add: reflp_def transp_def)
have "object_ptr_kinds h = object_ptr_kinds h2"
using h2
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
have "h2 ⊢ ok (set_child_nodes ptr (remove1 child children))"
proof (cases "is_element_ptr_kind ptr")
case True
then show ?thesis
using set_child_nodes_element_ok ‹known_ptr ptr› ‹object_ptr_kinds h = object_ptr_kinds h2›
‹type_wf h2› assms(4)
using ‹ptr |∈| object_ptr_kinds h› by blast
next
case False
then have "is_document_ptr_kind ptr"
using ‹known_ptr ptr› ‹ptr |∈| object_ptr_kinds h› ‹¬is_character_data_ptr ptr›
by(auto simp add:known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
moreover have "is_document_ptr ptr"
using ‹known_ptr ptr› ‹ptr |∈| object_ptr_kinds h› False ‹¬is_character_data_ptr ptr›
by(auto simp add: known_ptr_impl known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
ultimately show ?thesis
using assms(4)
apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def)[1]
apply(split invoke_splits)+
apply(auto elim!: bind_returns_result_E2 split: option.splits)[1]
apply(auto simp add: get_child_nodes⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r_def elim!: bind_returns_result_E2 split: option.splits)[1]
using assms(5) apply auto[1]
using ‹is_document_ptr_kind ptr› ‹known_ptr ptr› ‹object_ptr_kinds h = object_ptr_kinds h2›
‹ptr |∈| object_ptr_kinds h› ‹type_wf h2› local.set_child_nodes_document1_ok apply blast
using ‹is_document_ptr_kind ptr› ‹known_ptr ptr› ‹object_ptr_kinds h = object_ptr_kinds h2›
‹ptr |∈| object_ptr_kinds h› ‹type_wf h2› is_element_ptr_kind_cast local.set_child_nodes_document2_ok
apply blast
using ‹¬ is_character_data_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ptr› apply blast
by (metis False is_element_ptr_implies_kind option.case_eq_if)
qed
then
obtain h' where
h': "h2 ⊢ set_child_nodes ptr (remove1 child children) →⇩h h'"
by auto
show ?thesis
using assms
apply(auto simp add: remove_child_def
simp add: is_OK_returns_heap_I[OF h2] is_OK_returns_heap_I[OF h']
is_OK_returns_result_I[OF assms(4)] is_OK_returns_result_I[OF owner_document]
is_OK_returns_result_I[OF disconnected_nodes_h]
intro!: bind_is_OK_pure_I[OF get_owner_document_pure]
bind_is_OK_pure_I[OF get_child_nodes_pure]
bind_is_OK_pure_I[OF get_disconnected_nodes_pure]
bind_is_OK_I[rotated, OF h2]
dest!: returns_result_eq[OF assms(4)] returns_result_eq[OF owner_document]
returns_result_eq[OF disconnected_nodes_h]
)[1]
using h2 returns_result_select_result by force
qed
lemma adopt_node_ok:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "document_ptr |∈| document_ptr_kinds h"
assumes "child |∈| node_ptr_kinds h"
shows "h ⊢ ok (adopt_node document_ptr child)"
proof -
obtain old_document where
old_document: "h ⊢ get_owner_document (cast child) →⇩r old_document"
by (meson assms(1) assms(2) assms(3) assms(5) is_OK_returns_result_E local.get_owner_document_ok
node_ptr_kinds_commutes)
then have "h ⊢ ok (get_owner_document (cast child))"
by auto
obtain parent_opt where
parent_opt: "h ⊢ get_parent child →⇩r parent_opt"
by (meson assms(2) assms(3) is_OK_returns_result_I l_get_owner_document.get_owner_document_ptr_in_heap
local.get_parent_ok local.l_get_owner_document_axioms node_ptr_kinds_commutes old_document
returns_result_select_result)
then have "h ⊢ ok (get_parent child)"
by auto
have "h ⊢ ok (case parent_opt of Some parent ⇒ remove_child parent child | None ⇒ return ())"
apply(auto split: option.splits)[1]
using remove_child_ok
by (metis assms(1) assms(2) assms(3) local.get_parent_child_dual parent_opt)
then
obtain h2 where
h2: "h ⊢ (case parent_opt of Some parent ⇒ remove_child parent child | None ⇒ return ()) →⇩h h2"
by auto
have "object_ptr_kinds h = object_ptr_kinds h2"
using h2 apply(simp split: option.splits)
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes])
using remove_child_pointers_preserved
by (auto simp add: reflp_def transp_def)
then
have "old_document |∈| document_ptr_kinds h2"
using assms(1) assms(2) assms(3) document_ptr_kinds_commutes
local.get_owner_document_owner_document_in_heap old_document
by blast
have wellformed_h2: "heap_is_wellformed h2"
using h2 remove_child_heap_is_wellformed_preserved assms
by(auto split: option.splits)
have "type_wf h2"
using h2 remove_child_preserves_type_wf assms
by(auto split: option.splits)
have "known_ptrs h2"
using h2 remove_child_preserves_known_ptrs assms
by(auto split: option.splits)
have "object_ptr_kinds h = object_ptr_kinds h2"
using h2 apply(simp split: option.splits)
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF remove_child_writes])
using remove_child_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have "document_ptr_kinds h = document_ptr_kinds h2"
by(auto simp add: document_ptr_kinds_def)
have "h2 ⊢ ok (if document_ptr ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 child old_disc_nodes);
disc_nodes ← get_disconnected_nodes document_ptr;
set_disconnected_nodes document_ptr (child # disc_nodes)
} else do {
return ()
})"
proof(cases "document_ptr = old_document")
case True
then show ?thesis
by simp
next
case False
then have "h2 ⊢ ok (get_disconnected_nodes old_document)"
by (simp add: ‹old_document |∈| document_ptr_kinds h2› ‹type_wf h2› local.get_disconnected_nodes_ok)
then obtain old_disc_nodes where
old_disc_nodes: "h2 ⊢ get_disconnected_nodes old_document →⇩r old_disc_nodes"
by auto
have "h2 ⊢ ok (set_disconnected_nodes old_document (remove1 child old_disc_nodes))"
by (simp add: ‹old_document |∈| document_ptr_kinds h2› ‹type_wf h2› local.set_disconnected_nodes_ok)
then obtain h3 where
h3: "h2 ⊢ set_disconnected_nodes old_document (remove1 child old_disc_nodes) →⇩h h3"
by auto
have object_ptr_kinds_h2_eq3: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have object_ptr_kinds_M_eq_h2:
"⋀ptrs. h2 ⊢ object_ptr_kinds_M →⇩r ptrs = h3 ⊢ object_ptr_kinds_M →⇩r ptrs"
by(simp add: object_ptr_kinds_M_defs)
then have object_ptr_kinds_eq_h2: "|h2 ⊢ object_ptr_kinds_M|⇩r = |h3 ⊢ object_ptr_kinds_M|⇩r"
by(simp)
then have node_ptr_kinds_eq_h2: "|h2 ⊢ node_ptr_kinds_M|⇩r = |h3 ⊢ node_ptr_kinds_M|⇩r"
using node_ptr_kinds_M_eq by blast
then have node_ptr_kinds_eq3_h2: "node_ptr_kinds h2 = node_ptr_kinds h3"
by auto
have document_ptr_kinds_eq2_h2: "|h2 ⊢ document_ptr_kinds_M|⇩r = |h3 ⊢ document_ptr_kinds_M|⇩r"
using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
then have document_ptr_kinds_eq3_h2: "document_ptr_kinds h2 = document_ptr_kinds h3"
using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
have children_eq_h2:
"⋀ptr children. h2 ⊢ get_child_nodes ptr →⇩r children = h3 ⊢ get_child_nodes ptr →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h3
apply(rule reads_writes_preserved)
by (simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h2: "⋀ptr. |h2 ⊢ get_child_nodes ptr|⇩r = |h3 ⊢ get_child_nodes ptr|⇩r"
using select_result_eq by force
have "type_wf h3"
using ‹type_wf h2›
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
moreover have "document_ptr |∈| document_ptr_kinds h3"
using ‹document_ptr_kinds h = document_ptr_kinds h2› assms(4) document_ptr_kinds_eq3_h2 by auto
ultimately have "h3 ⊢ ok (get_disconnected_nodes document_ptr)"
by (simp add: local.get_disconnected_nodes_ok)
then obtain disc_nodes where
disc_nodes: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes"
by auto
have "h3 ⊢ ok (set_disconnected_nodes document_ptr (child # disc_nodes))"
using ‹document_ptr |∈| document_ptr_kinds h3› ‹type_wf h3› local.set_disconnected_nodes_ok by auto
then obtain h' where
h': "h3 ⊢ set_disconnected_nodes document_ptr (child # disc_nodes) →⇩h h'"
by auto
then show ?thesis
using False
using ‹h2 ⊢ ok get_disconnected_nodes old_document›
using ‹h3 ⊢ ok get_disconnected_nodes document_ptr›
apply(auto dest!: returns_result_eq[OF old_disc_nodes] returns_result_eq[OF disc_nodes]
intro!: bind_is_OK_I[rotated, OF h3] bind_is_OK_pure_I[OF get_disconnected_nodes_pure] )[1]
using ‹h2 ⊢ ok set_disconnected_nodes old_document (remove1 child old_disc_nodes)› by auto
qed
then obtain h' where
h': "h2 ⊢ (if document_ptr ≠ old_document then do {
old_disc_nodes ← get_disconnected_nodes old_document;
set_disconnected_nodes old_document (remove1 child old_disc_nodes);
disc_nodes ← get_disconnected_nodes document_ptr;
set_disconnected_nodes document_ptr (child # disc_nodes)
} else do {
return ()
}) →⇩h h'"
by auto
show ?thesis
using ‹h ⊢ ok (get_owner_document (cast child))›
using ‹h ⊢ ok (get_parent child)›
using h2 h'
apply(auto simp add: adopt_node_def
simp add: is_OK_returns_heap_I[OF h2]
intro!: bind_is_OK_pure_I[OF get_owner_document_pure]
bind_is_OK_pure_I[OF get_parent_pure]
bind_is_OK_I[rotated, OF h2]
dest!: returns_result_eq[OF parent_opt] returns_result_eq[OF old_document])[1]
using ‹h ⊢ ok (case parent_opt of None ⇒ return () | Some parent ⇒ remove_child parent child)›
by auto
qed
lemma insert_node_ok:
assumes "known_ptr parent" and "type_wf h"
assumes "parent |∈| object_ptr_kinds h"
assumes "¬is_character_data_ptr_kind parent"
assumes "is_document_ptr parent ⟹ h ⊢ get_child_nodes parent →⇩r []"
assumes "is_document_ptr parent ⟹ ¬is_character_data_ptr_kind node"
assumes "known_ptr (cast node)"
shows "h ⊢ ok (a_insert_node parent node ref)"
proof(auto simp add: a_insert_node_def get_child_nodes_ok[OF assms(1) assms(2) assms(3)]
intro!: bind_is_OK_pure_I)
fix children'
assume "h ⊢ get_child_nodes parent →⇩r children'"
show "h ⊢ ok set_child_nodes parent (insert_before_list node ref children')"
proof (cases "is_element_ptr_kind parent")
case True
then show ?thesis
using set_child_nodes_element_ok
using assms(1) assms(2) assms(3) by blast
next
case False
then have "is_document_ptr_kind parent"
using assms(4) assms(1)
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then have "is_document_ptr parent"
using assms(1)
by(auto simp add: known_ptr_impl DocumentClass.known_ptr_defs CharacterDataClass.known_ptr_defs
ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: option.splits)
then obtain children where children: "h ⊢ get_child_nodes parent →⇩r children" and "children = []"
using assms(5) by blast
have "insert_before_list node ref children' = [node]"
by (metis ‹children = []› ‹h ⊢ get_child_nodes parent →⇩r children'› append.left_neutral
children insert_Nil l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.insert_before_list.elims
l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs.insert_before_list.simps(3) neq_Nil_conv returns_result_eq)
moreover have "¬is_character_data_ptr_kind node"
using ‹is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r parent› assms(6) by blast
then have "is_element_ptr_kind node"
by (metis (no_types, lifting) CharacterDataClass.a_known_ptr_def DocumentClass.a_known_ptr_def
ElementClass.a_known_ptr_def NodeClass.a_known_ptr_def assms(7) cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r_inject
document_ptr_no_node_ptr_cast is_character_data_ptr_kind_none is_document_ptr_kind_none
is_element_ptr_implies_kind is_node_ptr_kind_cast local.known_ptr_impl node_ptr_casts_commute3
option.case_eq_if)
ultimately
show ?thesis
using set_child_nodes_document2_ok
by (metis ‹is_document_ptr⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r parent› assms(1) assms(2) assms(3) assms(5)
is_document_ptr_kind_none option.case_eq_if)
qed
qed
lemma insert_before_ok:
assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
assumes "parent |∈| object_ptr_kinds h"
assumes "node |∈| node_ptr_kinds h"
assumes "¬is_character_data_ptr_kind parent"
assumes "cast node ∉ set |h ⊢ get_ancestors parent|⇩r"
assumes "h ⊢ get_parent ref →⇩r Some parent"
assumes "is_document_ptr parent ⟹ h ⊢ get_child_nodes parent →⇩r []"
assumes "is_document_ptr parent ⟹ ¬is_character_data_ptr_kind node"
shows "h ⊢ ok (insert_before parent node (Some ref))"
proof -
have "h ⊢ ok (a_ensure_pre_insertion_validity node parent (Some ref))"
using assms ensure_pre_insertion_validity_ok by blast
have "h ⊢ ok (if Some node = Some ref
then a_next_sibling node
else return (Some ref))" (is "h ⊢ ok ?P")
apply(auto split: if_splits)[1]
using assms(1) assms(2) assms(3) assms(5) next_sibling_ok by blast
then obtain reference_child where
reference_child: "h ⊢ ?P →⇩r reference_child"
by auto
obtain owner_document where
owner_document: "h ⊢ get_owner_document parent →⇩r owner_document"
using assms get_owner_document_ok
by (meson returns_result_select_result)
then have "h ⊢ ok (get_owner_document parent)"
by auto
have "owner_document |∈| document_ptr_kinds h"
using assms(1) assms(2) assms(3) local.get_owner_document_owner_document_in_heap owner_document
by blast
obtain h2 where
h2: "h ⊢ adopt_node owner_document node →⇩h h2"
by (meson assms(1) assms(2) assms(3) assms(5) is_OK_returns_heap_E adopt_node_ok
l_insert_before_wf2⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms
local.get_owner_document_owner_document_in_heap owner_document)
then have "h ⊢ ok (adopt_node owner_document node)"
by auto
have "object_ptr_kinds h = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF adopt_node_writes h2])
using adopt_node_pointers_preserved
apply blast
by (auto simp add: reflp_def transp_def)
then have "document_ptr_kinds h = document_ptr_kinds h2"
by(auto simp add: document_ptr_kinds_def)
have "heap_is_wellformed h2"
using h2 adopt_node_preserves_wellformedness assms by blast
have "known_ptrs h2"
using h2 adopt_node_preserves_known_ptrs assms by blast
have "type_wf h2"
using h2 adopt_node_preserves_type_wf assms by blast
obtain disconnected_nodes_h2 where
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h2"
by (metis ‹document_ptr_kinds h = document_ptr_kinds h2› ‹type_wf h2› assms(1) assms(2) assms(3)
is_OK_returns_result_E local.get_disconnected_nodes_ok local.get_owner_document_owner_document_in_heap
owner_document)
obtain h3 where
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) →⇩h h3"
by (metis ‹document_ptr_kinds h = document_ptr_kinds h2› ‹owner_document |∈| document_ptr_kinds h›
‹type_wf h2› document_ptr_kinds_def is_OK_returns_heap_E
l_set_disconnected_nodes.set_disconnected_nodes_ok local.l_set_disconnected_nodes_axioms)
have "type_wf h3"
using ‹type_wf h2›
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
OF set_disconnected_nodes_writes h3])
unfolding a_remove_child_locs_def
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
have "parent |∈| object_ptr_kinds h3"
using ‹object_ptr_kinds h = object_ptr_kinds h2› assms(4) object_ptr_kinds_M_eq3_h2 by blast
moreover have "known_ptr parent"
using assms(2) assms(4) local.known_ptrs_known_ptr by blast
moreover have "known_ptr (cast node)"
using assms(2) assms(5) local.known_ptrs_known_ptr node_ptr_kinds_commutes by blast
moreover have "is_document_ptr parent ⟹ h3 ⊢ get_child_nodes parent →⇩r []"
by (metis assms(8) assms(9) distinct.simps(2) distinct_singleton local.get_parent_child_dual
returns_result_eq)
ultimately obtain h' where
h': "h3 ⊢ a_insert_node parent node reference_child →⇩h h'"
using insert_node_ok ‹type_wf h3› assms by blast
show ?thesis
using ‹h ⊢ ok (a_ensure_pre_insertion_validity node parent (Some ref))›
using reference_child ‹h ⊢ ok (get_owner_document parent)› ‹h ⊢ ok (adopt_node owner_document node)›
h3 h'
apply(auto simp add: insert_before_def
simp add: is_OK_returns_result_I[OF disconnected_nodes_h2]
simp add: is_OK_returns_heap_I[OF h3] is_OK_returns_heap_I[OF h']
intro!: bind_is_OK_I2
bind_is_OK_pure_I[OF ensure_pre_insertion_validity_pure]
bind_is_OK_pure_I[OF next_sibling_pure]
bind_is_OK_pure_I[OF get_owner_document_pure]
bind_is_OK_pure_I[OF get_disconnected_nodes_pure]
dest!: returns_result_eq[OF owner_document] returns_result_eq[OF disconnected_nodes_h2]
returns_heap_eq[OF h2] returns_heap_eq[OF h3]
dest!: sym[of node ref]
)[1]
using returns_result_eq by fastforce
qed
end
interpretation i_insert_before_wf3?: l_insert_before_wf3⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs
get_ancestors get_ancestors_locs adopt_node adopt_node_locs set_disconnected_nodes
set_disconnected_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_owner_document
insert_before insert_before_locs append_child type_wf known_ptr known_ptrs heap_is_wellformed
parent_child_rel remove_child remove_child_locs get_root_node get_root_node_locs remove
by(auto simp add: l_insert_before_wf3⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_insert_before_wf3⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
locale l_append_child_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_adopt_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_insert_before⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_append_child⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_insert_before_wf +
l_insert_before_wf2 +
l_get_child_nodes
begin
lemma append_child_heap_is_wellformed_preserved:
assumes wellformed: "heap_is_wellformed h"
and append_child: "h ⊢ append_child ptr node →⇩h h'"
and known_ptrs: "known_ptrs h"
and type_wf: "type_wf h"
shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs h'"
using assms
by(auto simp add: append_child_def intro: insert_before_preserves_type_wf
insert_before_preserves_known_ptrs insert_before_heap_is_wellformed_preserved)
lemma append_child_children:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r xs"
assumes "h ⊢ append_child ptr node →⇩h h'"
assumes "node ∉ set xs"
shows "h' ⊢ get_child_nodes ptr →⇩r xs @ [node]"
proof -
obtain ancestors owner_document h2 h3 disconnected_nodes_h2 where
ancestors: "h ⊢ get_ancestors ptr →⇩r ancestors" and
node_not_in_ancestors: "cast node ∉ set ancestors" and
owner_document: "h ⊢ get_owner_document ptr →⇩r owner_document" and
h2: "h ⊢ adopt_node owner_document node →⇩h h2" and
disconnected_nodes_h2: "h2 ⊢ get_disconnected_nodes owner_document →⇩r disconnected_nodes_h2" and
h3: "h2 ⊢ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) →⇩h h3" and
h': "h3 ⊢ a_insert_node ptr node None →⇩h h'"
using assms(5)
by(auto simp add: append_child_def insert_before_def a_ensure_pre_insertion_validity_def
elim!: bind_returns_heap_E bind_returns_result_E
bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
split: if_splits option.splits)
have "⋀parent. |h ⊢ get_parent node|⇩r = Some parent ⟹ parent ≠ ptr"
using assms(1) assms(4) assms(6)
by (metis (no_types, lifting) assms(2) assms(3) h2 is_OK_returns_heap_I is_OK_returns_result_E
local.adopt_node_child_in_heap local.get_parent_child_dual local.get_parent_ok
select_result_I2)
have "h2 ⊢ get_child_nodes ptr →⇩r xs"
using get_child_nodes_reads adopt_node_writes h2 assms(4)
apply(rule reads_writes_separate_forwards)
using ‹⋀parent. |h ⊢ get_parent node|⇩r = Some parent ⟹ parent ≠ ptr›
apply(auto simp add: adopt_node_locs_def remove_child_locs_def)[1]
by (meson local.set_child_nodes_get_child_nodes_different_pointers)
have "h3 ⊢ get_child_nodes ptr →⇩r xs"
using get_child_nodes_reads set_disconnected_nodes_writes h3 ‹h2 ⊢ get_child_nodes ptr →⇩r xs›
apply(rule reads_writes_separate_forwards)
by(auto)
have "ptr |∈| object_ptr_kinds h"
by (meson ancestors is_OK_returns_result_I local.get_ancestors_ptr_in_heap)
then
have "known_ptr ptr"
using assms(3)
using local.known_ptrs_known_ptr by blast
have "type_wf h2"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF adopt_node_writes h2]
using adopt_node_types_preserved ‹type_wf h›
by(auto simp add: adopt_node_locs_def remove_child_locs_def reflp_def transp_def split: if_splits)
then
have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
show "h' ⊢ get_child_nodes ptr →⇩r xs@[node]"
using h'
apply(auto simp add: a_insert_node_def
dest!: bind_returns_heap_E3[rotated, OF ‹h3 ⊢ get_child_nodes ptr →⇩r xs›
get_child_nodes_pure, rotated])[1]
using ‹type_wf h3› set_child_nodes_get_child_nodes ‹known_ptr ptr›
by metis
qed
lemma append_child_for_all_on_children:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r xs"
assumes "h ⊢ forall_M (append_child ptr) nodes →⇩h h'"
assumes "set nodes ∩ set xs = {}"
assumes "distinct nodes"
shows "h' ⊢ get_child_nodes ptr →⇩r xs@nodes"
using assms
apply(induct nodes arbitrary: h xs)
apply(simp)
proof(auto elim!: bind_returns_heap_E)[1]fix a nodes h xs h'a
assume 0: "(⋀h xs. heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h
⟹ h ⊢ get_child_nodes ptr →⇩r xs ⟹ h ⊢ forall_M (append_child ptr) nodes →⇩h h'
⟹ set nodes ∩ set xs = {} ⟹ h' ⊢ get_child_nodes ptr →⇩r xs @ nodes)"
and 1: "heap_is_wellformed h"
and 2: "type_wf h"
and 3: "known_ptrs h"
and 4: "h ⊢ get_child_nodes ptr →⇩r xs"
and 5: "h ⊢ append_child ptr a →⇩r ()"
and 6: "h ⊢ append_child ptr a →⇩h h'a"
and 7: "h'a ⊢ forall_M (append_child ptr) nodes →⇩h h'"
and 8: "a ∉ set xs"
and 9: "set nodes ∩ set xs = {}"
and 10: "a ∉ set nodes"
and 11: "distinct nodes"
then have "h'a ⊢ get_child_nodes ptr →⇩r xs @ [a]"
using append_child_children 6
using "1" "2" "3" "4" "8" by blast
moreover have "heap_is_wellformed h'a" and "type_wf h'a" and "known_ptrs h'a"
using insert_before_heap_is_wellformed_preserved insert_before_preserves_known_ptrs
insert_before_preserves_type_wf 1 2 3 6 append_child_def
by metis+
moreover have "set nodes ∩ set (xs @ [a]) = {}"
using 9 10
by auto
ultimately show "h' ⊢ get_child_nodes ptr →⇩r xs @ a # nodes"
using 0 7
by fastforce
qed
lemma append_child_for_all_on_no_children:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r []"
assumes "h ⊢ forall_M (append_child ptr) nodes →⇩h h'"
assumes "distinct nodes"
shows "h' ⊢ get_child_nodes ptr →⇩r nodes"
using assms append_child_for_all_on_children
by force
end
locale l_append_child_wf = l_type_wf + l_known_ptrs + l_append_child_defs + l_heap_is_wellformed_defs +
assumes append_child_preserves_type_wf:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ append_child ptr child →⇩h h'
⟹ type_wf h'"
assumes append_child_preserves_known_ptrs:
"heap_is_wellformed h ⟹ type_wf h ⟹ known_ptrs h ⟹ h ⊢ append_child ptr child →⇩h h'
⟹ known_ptrs h'"
assumes append_child_heap_is_wellformed_preserved:
"type_wf h ⟹ known_ptrs h ⟹ heap_is_wellformed h ⟹ h ⊢ append_child ptr child →⇩h h'
⟹ heap_is_wellformed h'"
interpretation i_append_child_wf?: l_append_child_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_owner_document get_parent
get_parent_locs remove_child remove_child_locs
get_disconnected_nodes get_disconnected_nodes_locs
set_disconnected_nodes set_disconnected_nodes_locs
adopt_node adopt_node_locs known_ptr type_wf get_child_nodes
get_child_nodes_locs known_ptrs set_child_nodes
set_child_nodes_locs remove get_ancestors get_ancestors_locs
insert_before insert_before_locs append_child heap_is_wellformed
parent_child_rel
by(auto simp add: l_append_child_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
lemma append_child_wf_is_l_append_child_wf [instances]: "l_append_child_wf type_wf known_ptr
known_ptrs append_child heap_is_wellformed"
apply(auto simp add: l_append_child_wf_def l_append_child_wf_axioms_def instances)[1]
using append_child_heap_is_wellformed_preserved by fast+
subsection ‹create\_element›
locale l_create_element_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs
heap_is_wellformed parent_child_rel +
l_new_element_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs +
l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs
get_disconnected_nodes get_disconnected_nodes_locs +
l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs set_tag_name set_tag_name_locs type_wf create_element known_ptr +
l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs +
l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr
get_child_nodes get_child_nodes_locs +
l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes set_disconnected_nodes_locs
get_child_nodes get_child_nodes_locs +
l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs +
l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs +
l_new_element type_wf +
l_known_ptrs known_ptr known_ptrs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and set_tag_name :: "(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and create_element :: "(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr) prog"
begin
lemma create_element_preserves_wellformedness:
assumes "heap_is_wellformed h"
and "h ⊢ create_element document_ptr tag →⇩h h'"
and "type_wf h"
and "known_ptrs h"
shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs h'"
proof -
obtain new_element_ptr h2 h3 disc_nodes_h3 where
new_element_ptr: "h ⊢ new_element →⇩r new_element_ptr" and
h2: "h ⊢ new_element →⇩h h2" and
h3: "h2 ⊢ set_tag_name new_element_ptr tag →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes_h3) →⇩h h'"
using assms(2)
by(auto simp add: create_element_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
then have "h ⊢ create_element document_ptr tag →⇩r new_element_ptr"
apply(auto simp add: create_element_def intro!: bind_returns_result_I)[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
apply (metis is_OK_returns_heap_E is_OK_returns_result_I local.get_disconnected_nodes_pure
pure_returns_heap_eq)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "new_element_ptr ∉ set |h ⊢ element_ptr_kinds_M|⇩r"
using new_element_ptr ElementMonad.ptr_kinds_ptr_kinds_M h2
using new_element_ptr_not_in_heap by blast
then have "cast new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r"
by simp
then have "cast new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r"
by simp
have object_ptr_kinds_eq_h: "object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_element_ptr|}"
using new_element_new_ptr h2 new_element_ptr by blast
then have node_ptr_kinds_eq_h: "node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_element_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h |∪| {|new_element_ptr|}"
apply(simp add: element_ptr_kinds_def)
by force
have character_data_ptr_kinds_eq_h: "character_data_ptr_kinds h2 = character_data_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def character_data_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_tag_name_writes h3])
using set_tag_name_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "known_ptr (cast new_element_ptr)"
using ‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr› local.create_element_known_ptr
by blast
then
have "known_ptrs h2"
using known_ptrs_new_ptr object_ptr_kinds_eq_h ‹known_ptrs h› h2
by blast
then
have "known_ptrs h3"
using known_ptrs_preserved object_ptr_kinds_eq_h2 by blast
then
show "known_ptrs h'"
using known_ptrs_preserved object_ptr_kinds_eq_h3 by blast
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_element_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2 get_child_nodes_new_element[rotated, OF new_element_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h: "⋀ptr'. ptr' ≠ cast new_element_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []"
using new_element_ptr h2 new_element_ptr_in_heap[OF h2 new_element_ptr]
new_element_is_element_ptr[OF new_element_ptr] new_element_no_child_nodes
by blast
have disconnected_nodes_eq_h:
"⋀doc_ptr disc_nodes. h ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads h2 get_disconnected_nodes_new_element[OF new_element_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have disconnected_nodes_eq2_h:
"⋀doc_ptr. |h ⊢ get_disconnected_nodes doc_ptr|⇩r = |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_tag_name_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_tag_name_get_child_nodes)
then have children_eq2_h2: "⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_tag_name_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_tag_name_get_disconnected_nodes)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have "type_wf h2"
using ‹type_wf h› new_element_types_preserved h2 by blast
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_tag_name_writes h3]
using set_tag_name_types_preserved
by(auto simp add: reflp_def transp_def)
then show "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_eq_h3:
"⋀ptr' children. h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h3: "⋀ptr'. |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. document_ptr ≠ doc_ptr
⟹ h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disc_nodes_document_ptr_h2: "h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h2 disc_nodes_h3 by auto
then have disc_nodes_document_ptr_h: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h by auto
then have "cast new_element_ptr ∉ set disc_nodes_h3"
using ‹heap_is_wellformed h›
using ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
a_all_ptrs_in_heap_def heap_is_wellformed_def
using NodeMonad.ptr_kinds_ptr_kinds_M local.heap_is_wellformed_disc_nodes_in_heap by blast
have "acyclic (parent_child_rel h)"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def acyclic_heap_def)
also have "parent_child_rel h = parent_child_rel h2"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h2"
by (simp add: object_ptr_kinds_eq_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq_h ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting)
‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []›
children_eq2_h empty_iff empty_set image_eqI select_result_I2)
qed
also have "… = parent_child_rel h3"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
also have "… = parent_child_rel h'"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
finally have "a_acyclic_heap h'"
by (simp add: acyclic_heap_def)
have "a_all_ptrs_in_heap h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h2"
apply(auto simp add: a_all_ptrs_in_heap_def)[1]
apply (metis ‹known_ptrs h2› ‹parent_child_rel h = parent_child_rel h2› ‹type_wf h2› assms(1)
assms(3) funion_iff local.get_child_nodes_ok local.known_ptrs_known_ptr
local.parent_child_rel_child_in_heap local.parent_child_rel_child_nodes2 node_ptr_kinds_commutes
node_ptr_kinds_eq_h returns_result_select_result)
by (metis assms(1) assms(3) disconnected_nodes_eq2_h document_ptr_kinds_eq_h funion_iff
local.get_disconnected_nodes_ok local.heap_is_wellformed_disc_nodes_in_heap node_ptr_kinds_eq_h
returns_result_select_result)
then have "a_all_ptrs_in_heap h3"
by (simp add: children_eq2_h2 disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
local.a_all_ptrs_in_heap_def node_ptr_kinds_eq_h2 object_ptr_kinds_eq_h2)
then have "a_all_ptrs_in_heap h'"
by (smt ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []› children_eq2_h3
disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h3
finite_set_in h' is_OK_returns_result_I set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.get_child_nodes_ptr_in_heap node_ptr_kinds_commutes
object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 select_result_I2 set_ConsD subset_code(1))
have "⋀p. p |∈| object_ptr_kinds h ⟹ cast new_element_ptr ∉ set |h ⊢ get_child_nodes p|⇩r"
using ‹heap_is_wellformed h› ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
heap_is_wellformed_children_in_heap
by (meson NodeMonad.ptr_kinds_ptr_kinds_M a_all_ptrs_in_heap_def assms(3) assms(4) fset_mp
fset_of_list_elem get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result)
then have "⋀p. p |∈| object_ptr_kinds h2 ⟹ cast new_element_ptr ∉ set |h2 ⊢ get_child_nodes p|⇩r"
using children_eq2_h
apply(auto simp add: object_ptr_kinds_eq_h)[1]
using ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []› apply auto[1]
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›)
then have "⋀p. p |∈| object_ptr_kinds h3 ⟹ cast new_element_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h2 children_eq2_h2 by auto
then have new_element_ptr_not_in_any_children:
"⋀p. p |∈| object_ptr_kinds h' ⟹ cast new_element_ptr ∉ set |h' ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h3 children_eq2_h3 by auto
have "a_distinct_lists h"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h2"
using ‹h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []›
apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h
disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1]
apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert)
apply(case_tac "x=cast new_element_ptr")
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply (metis IntI assms(1) assms(3) assms(4) empty_iff local.get_child_nodes_ok
local.heap_is_wellformed_one_parent local.known_ptrs_known_ptr returns_result_select_result)
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
by (metis ‹local.a_distinct_lists h› ‹type_wf h2› disconnected_nodes_eq_h document_ptr_kinds_eq_h
local.distinct_lists_no_parent local.get_disconnected_nodes_ok returns_result_select_result)
then have "a_distinct_lists h3"
by(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
children_eq2_h2 object_ptr_kinds_eq_h2)
then have "a_distinct_lists h'"
proof(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h3 children_eq2_h3
object_ptr_kinds_eq_h3 document_ptr_kinds_eq_h3
intro!: distinct_concat_map_I)[1]
fix x
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
then show "distinct |h' ⊢ get_disconnected_nodes x|⇩r"
using document_ptr_kinds_eq_h3 disconnected_nodes_eq_h3 h' set_disconnected_nodes_get_disconnected_nodes
by (metis (no_types, lifting) ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set disc_nodes_h3›
‹a_distinct_lists h3› ‹type_wf h'› disc_nodes_h3 distinct.simps(2)
distinct_lists_disconnected_nodes get_disconnected_nodes_ok returns_result_eq
returns_result_select_result)
next
fix x y xa
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
and "y |∈| document_ptr_kinds h3"
and "x ≠ y"
and "xa ∈ set |h' ⊢ get_disconnected_nodes x|⇩r"
and "xa ∈ set |h' ⊢ get_disconnected_nodes y|⇩r"
moreover have "set |h3 ⊢ get_disconnected_nodes x|⇩r ∩ set |h3 ⊢ get_disconnected_nodes y|⇩r = {}"
using calculation by(auto dest: distinct_concat_map_E(1))
ultimately show "False"
apply(-)
apply(cases "x = document_ptr")
apply (smt NodeMonad.ptr_kinds_ptr_kinds_M ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
‹local.a_all_ptrs_in_heap h›
disc_nodes_h3 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3
disjoint_iff_not_equal document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
select_result_I2 set_ConsD subsetD)
by (smt NodeMonad.ptr_kinds_ptr_kinds_M ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
‹local.a_all_ptrs_in_heap h›
disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3
disjoint_iff_not_equal document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
select_result_I2 set_ConsD subsetD)
next
fix x xa xb
assume 2: "(⋃x∈fset (object_ptr_kinds h3). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h3). set |h3 ⊢ get_disconnected_nodes x|⇩r) = {}"
and 3: "xa |∈| object_ptr_kinds h3"
and 4: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 5: "xb |∈| document_ptr_kinds h3"
and 6: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
show "False"
using disc_nodes_document_ptr_h disconnected_nodes_eq2_h3
apply -
apply(cases "xb = document_ptr")
apply (metis (no_types, hide_lams) "3" "4" "6"
‹⋀p. p |∈| object_ptr_kinds h3
⟹ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r›
‹a_distinct_lists h3› children_eq2_h3 disc_nodes_h3 distinct_lists_no_parent h'
select_result_I2 set_ConsD set_disconnected_nodes_get_disconnected_nodes)
by (metis "3" "4" "5" "6" ‹a_distinct_lists h3› ‹type_wf h3› children_eq2_h3
distinct_lists_no_parent get_disconnected_nodes_ok returns_result_select_result)
qed
have "a_owner_document_valid h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
using disc_nodes_h3 ‹document_ptr |∈| document_ptr_kinds h›
apply(auto simp add: a_owner_document_valid_def)[1]
apply(auto simp add: object_ptr_kinds_eq_h object_ptr_kinds_eq_h3 )[1]
apply(auto simp add: object_ptr_kinds_eq_h2)[1]
apply(auto simp add: document_ptr_kinds_eq_h document_ptr_kinds_eq_h3 )[1]
apply(auto simp add: document_ptr_kinds_eq_h2)[1]
apply(auto simp add: node_ptr_kinds_eq_h node_ptr_kinds_eq_h3 )[1]
apply(auto simp add: node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h )[1]
apply(auto simp add: children_eq2_h2[symmetric] children_eq2_h3[symmetric]
disconnected_nodes_eq2_h disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3)[1]
apply (metis (no_types, lifting) document_ptr_kinds_eq_h h' list.set_intros(1)
local.set_disconnected_nodes_get_disconnected_nodes select_result_I2)
apply(simp add: object_ptr_kinds_eq_h)
by (smt ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h
children_eq2_h2 children_eq2_h3 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h finite_set_in h' list.set_intros(2)
local.set_disconnected_nodes_get_disconnected_nodes select_result_I2)
show "heap_is_wellformed h'"
using ‹a_acyclic_heap h'› ‹a_all_ptrs_in_heap h'› ‹a_distinct_lists h'› ‹a_owner_document_valid h'›
by(simp add: heap_is_wellformed_def)
qed
end
interpretation i_create_element_wf?: l_create_element_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr known_ptrs type_wf
get_child_nodes get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs heap_is_wellformed parent_child_rel
set_tag_name set_tag_name_locs
set_disconnected_nodes set_disconnected_nodes_locs create_element
using instances
by(auto simp add: l_create_element_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_create_element_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsection ‹create\_character\_data›
locale l_create_character_data_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs heap_is_wellformed parent_child_rel
+ l_new_character_data_get_disconnected_nodes
get_disconnected_nodes get_disconnected_nodes_locs
+ l_set_val_get_disconnected_nodes
type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs
+ l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs set_val set_val_locs type_wf create_character_data known_ptr
+ l_new_character_data_get_child_nodes
type_wf known_ptr get_child_nodes get_child_nodes_locs
+ l_set_val_get_child_nodes
type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs
+ l_set_disconnected_nodes_get_child_nodes
set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs
+ l_set_disconnected_nodes
type_wf set_disconnected_nodes set_disconnected_nodes_locs
+ l_set_disconnected_nodes_get_disconnected_nodes
type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs
+ l_new_character_data
type_wf
+ l_known_ptrs
known_ptr known_ptrs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and set_val :: "(_) character_data_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_val_locs :: "(_) character_data_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_disconnected_nodes ::
"(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and create_character_data ::
"(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) character_data_ptr) prog"
and known_ptrs :: "(_) heap ⇒ bool"
begin
lemma create_character_data_preserves_wellformedness:
assumes "heap_is_wellformed h"
and "h ⊢ create_character_data document_ptr text →⇩h h'"
and "type_wf h"
and "known_ptrs h"
shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs h'"
proof -
obtain new_character_data_ptr h2 h3 disc_nodes_h3 where
new_character_data_ptr: "h ⊢ new_character_data →⇩r new_character_data_ptr" and
h2: "h ⊢ new_character_data →⇩h h2" and
h3: "h2 ⊢ set_val new_character_data_ptr text →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes_h3) →⇩h h'"
using assms(2)
by(auto simp add: create_character_data_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
then
have "h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr"
apply(auto simp add: create_character_data_def intro!: bind_returns_result_I)[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
apply (metis is_OK_returns_heap_E is_OK_returns_result_I local.get_disconnected_nodes_pure
pure_returns_heap_eq)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "new_character_data_ptr ∉ set |h ⊢ character_data_ptr_kinds_M|⇩r"
using new_character_data_ptr CharacterDataMonad.ptr_kinds_ptr_kinds_M h2
using new_character_data_ptr_not_in_heap by blast
then have "cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r"
by simp
then have "cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r"
by simp
have object_ptr_kinds_eq_h:
"object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using new_character_data_new_ptr h2 new_character_data_ptr by blast
then have node_ptr_kinds_eq_h:
"node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have character_data_ptr_kinds_eq_h:
"character_data_ptr_kinds h2 = character_data_ptr_kinds h |∪| {|new_character_data_ptr|}"
apply(simp add: character_data_ptr_kinds_def)
by force
have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_val_writes h3])
using set_val_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_character_data_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2
get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h:
"⋀ptr'. ptr' ≠ cast new_character_data_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have object_ptr_kinds_eq_h:
"object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using new_character_data_new_ptr h2 new_character_data_ptr by blast
then have node_ptr_kinds_eq_h:
"node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have character_data_ptr_kinds_eq_h:
"character_data_ptr_kinds h2 = character_data_ptr_kinds h |∪| {|new_character_data_ptr|}"
apply(simp add: character_data_ptr_kinds_def)
by force
have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_val_writes h3])
using set_val_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_character_data_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2 get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h: "⋀ptr'. ptr' ≠ cast new_character_data_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []"
using new_character_data_ptr h2 new_character_data_ptr_in_heap[OF h2 new_character_data_ptr]
new_character_data_is_character_data_ptr[OF new_character_data_ptr]
new_character_data_no_child_nodes
by blast
have disconnected_nodes_eq_h:
"⋀doc_ptr disc_nodes. h ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads h2
get_disconnected_nodes_new_character_data[OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have disconnected_nodes_eq2_h:
"⋀doc_ptr. |h ⊢ get_disconnected_nodes doc_ptr|⇩r = |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_val_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_val_get_child_nodes)
then have children_eq2_h2:
"⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_val_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_val_get_disconnected_nodes)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have "type_wf h2"
using ‹type_wf h› new_character_data_types_preserved h2 by blast
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_val_writes h3]
using set_val_types_preserved
by(auto simp add: reflp_def transp_def)
then show "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_eq_h3:
"⋀ptr' children. h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h3:
" ⋀ptr'. |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h3: "⋀doc_ptr disc_nodes. document_ptr ≠ doc_ptr
⟹ h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h3: "⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disc_nodes_document_ptr_h2: "h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h2 disc_nodes_h3 by auto
then have disc_nodes_document_ptr_h: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h by auto
then have "cast new_character_data_ptr ∉ set disc_nodes_h3"
using ‹heap_is_wellformed h› using ‹cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
a_all_ptrs_in_heap_def heap_is_wellformed_def
using NodeMonad.ptr_kinds_ptr_kinds_M local.heap_is_wellformed_disc_nodes_in_heap by blast
have "acyclic (parent_child_rel h)"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def acyclic_heap_def)
also have "parent_child_rel h = parent_child_rel h2"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h2"
by (simp add: object_ptr_kinds_eq_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq_h ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting) ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
children_eq2_h empty_iff empty_set image_eqI select_result_I2)
qed
also have "… = parent_child_rel h3"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
also have "… = parent_child_rel h'"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
finally have "a_acyclic_heap h'"
by (simp add: acyclic_heap_def)
have "a_all_ptrs_in_heap h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h2"
apply(auto simp add: a_all_ptrs_in_heap_def)[1]
using node_ptr_kinds_eq_h ‹cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
apply (metis (no_types, lifting) NodeMonad.ptr_kinds_ptr_kinds_M ‹parent_child_rel h = parent_child_rel h2›
children_eq2_h finite_set_in finsert_iff funion_finsert_right local.parent_child_rel_child
local.parent_child_rel_parent_in_heap node_ptr_kinds_commutes object_ptr_kinds_eq_h
select_result_I2 subsetD sup_bot.right_neutral)
by (metis assms(1) assms(3) disconnected_nodes_eq2_h document_ptr_kinds_eq_h funionI1
local.get_disconnected_nodes_ok local.heap_is_wellformed_disc_nodes_in_heap
node_ptr_kinds_eq_h returns_result_select_result)
then have "a_all_ptrs_in_heap h3"
by (simp add: children_eq2_h2 disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
local.a_all_ptrs_in_heap_def node_ptr_kinds_eq_h2 object_ptr_kinds_eq_h2)
then have "a_all_ptrs_in_heap h'"
by (smt character_data_ptr_kinds_commutes children_eq2_h3 disc_nodes_document_ptr_h2
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h3
finite_set_in h' h2 local.a_all_ptrs_in_heap_def
local.set_disconnected_nodes_get_disconnected_nodes new_character_data_ptr
new_character_data_ptr_in_heap node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h3
object_ptr_kinds_eq_h3 select_result_I2 set_ConsD subset_code(1))
have "⋀p. p |∈| object_ptr_kinds h ⟹ cast new_character_data_ptr ∉ set |h ⊢ get_child_nodes p|⇩r"
using ‹heap_is_wellformed h› ‹cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
heap_is_wellformed_children_in_heap
by (meson NodeMonad.ptr_kinds_ptr_kinds_M a_all_ptrs_in_heap_def assms(3) assms(4) fset_mp
fset_of_list_elem get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result)
then have "⋀p. p |∈| object_ptr_kinds h2 ⟹ cast new_character_data_ptr ∉ set |h2 ⊢ get_child_nodes p|⇩r"
using children_eq2_h
apply(auto simp add: object_ptr_kinds_eq_h)[1]
using ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []› apply auto[1]
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M ‹cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›)
then have "⋀p. p |∈| object_ptr_kinds h3 ⟹ cast new_character_data_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h2 children_eq2_h2 by auto
then have new_character_data_ptr_not_in_any_children:
"⋀p. p |∈| object_ptr_kinds h' ⟹ cast new_character_data_ptr ∉ set |h' ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h3 children_eq2_h3 by auto
have "a_distinct_lists h"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h2"
using ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h
disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1]
apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert)
apply(case_tac "x=cast new_character_data_ptr")
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply (metis IntI assms(1) assms(3) assms(4) empty_iff local.get_child_nodes_ok
local.heap_is_wellformed_one_parent local.known_ptrs_known_ptr
returns_result_select_result)
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
by (metis ‹local.a_distinct_lists h› ‹type_wf h2› disconnected_nodes_eq_h document_ptr_kinds_eq_h
local.distinct_lists_no_parent local.get_disconnected_nodes_ok returns_result_select_result)
then have "a_distinct_lists h3"
by(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
children_eq2_h2 object_ptr_kinds_eq_h2)[1]
then have "a_distinct_lists h'"
proof(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h3 children_eq2_h3
object_ptr_kinds_eq_h3 document_ptr_kinds_eq_h3 intro!: distinct_concat_map_I)[1]
fix x
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
then show "distinct |h' ⊢ get_disconnected_nodes x|⇩r"
using document_ptr_kinds_eq_h3 disconnected_nodes_eq_h3 h' set_disconnected_nodes_get_disconnected_nodes
by (metis (no_types, lifting) ‹cast new_character_data_ptr ∉ set disc_nodes_h3›
‹a_distinct_lists h3› ‹type_wf h'› disc_nodes_h3 distinct.simps(2)
distinct_lists_disconnected_nodes get_disconnected_nodes_ok returns_result_eq
returns_result_select_result)
next
fix x y xa
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
and "y |∈| document_ptr_kinds h3"
and "x ≠ y"
and "xa ∈ set |h' ⊢ get_disconnected_nodes x|⇩r"
and "xa ∈ set |h' ⊢ get_disconnected_nodes y|⇩r"
moreover have "set |h3 ⊢ get_disconnected_nodes x|⇩r ∩ set |h3 ⊢ get_disconnected_nodes y|⇩r = {}"
using calculation by(auto dest: distinct_concat_map_E(1))
ultimately show "False"
by (smt NodeMonad.ptr_kinds_ptr_kinds_M
‹cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
‹local.a_all_ptrs_in_heap h› disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 disjoint_iff_not_equal
document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
select_result_I2 set_ConsD subsetD)
next
fix x xa xb
assume 2: "(⋃x∈fset (object_ptr_kinds h3). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h3). set |h3 ⊢ get_disconnected_nodes x|⇩r) = {}"
and 3: "xa |∈| object_ptr_kinds h3"
and 4: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 5: "xb |∈| document_ptr_kinds h3"
and 6: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
show "False"
using disc_nodes_document_ptr_h disconnected_nodes_eq2_h3
apply(cases "xb = document_ptr")
apply (metis (no_types, hide_lams) "3" "4" "6"
‹⋀p. p |∈| object_ptr_kinds h3 ⟹ cast new_character_data_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r›
‹a_distinct_lists h3› children_eq2_h3 disc_nodes_h3 distinct_lists_no_parent h'
select_result_I2 set_ConsD set_disconnected_nodes_get_disconnected_nodes)
by (metis "3" "4" "5" "6" ‹a_distinct_lists h3› ‹type_wf h3› children_eq2_h3
distinct_lists_no_parent get_disconnected_nodes_ok returns_result_select_result)
qed
have "a_owner_document_valid h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
using disc_nodes_h3 ‹document_ptr |∈| document_ptr_kinds h›
apply(simp add: a_owner_document_valid_def)
apply(simp add: object_ptr_kinds_eq_h object_ptr_kinds_eq_h3 )
apply(simp add: object_ptr_kinds_eq_h2)
apply(simp add: document_ptr_kinds_eq_h document_ptr_kinds_eq_h3 )
apply(simp add: document_ptr_kinds_eq_h2)
apply(simp add: node_ptr_kinds_eq_h node_ptr_kinds_eq_h3 )
apply(simp add: node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h )
apply(auto simp add: children_eq2_h2[symmetric] children_eq2_h3[symmetric] disconnected_nodes_eq2_h
disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3)[1]
apply (metis (no_types, lifting) document_ptr_kinds_eq_h h' list.set_intros(1)
local.set_disconnected_nodes_get_disconnected_nodes select_result_I2)
apply(simp add: object_ptr_kinds_eq_h)
by (smt ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h
disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h finite_set_in h' list.set_intros(2)
local.set_disconnected_nodes_get_disconnected_nodes select_result_I2)
have "known_ptr (cast new_character_data_ptr)"
using ‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr›
local.create_character_data_known_ptr by blast
then
have "known_ptrs h2"
using known_ptrs_new_ptr object_ptr_kinds_eq_h ‹known_ptrs h› h2
by blast
then
have "known_ptrs h3"
using known_ptrs_preserved object_ptr_kinds_eq_h2 by blast
then
show "known_ptrs h'"
using known_ptrs_preserved object_ptr_kinds_eq_h3 by blast
show "heap_is_wellformed h'"
using ‹a_acyclic_heap h'› ‹a_all_ptrs_in_heap h'› ‹a_distinct_lists h'› ‹a_owner_document_valid h'›
by(simp add: heap_is_wellformed_def)
qed
end
interpretation i_create_character_data_wf?: l_create_character_data_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf
get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
heap_is_wellformed parent_child_rel set_val set_val_locs set_disconnected_nodes
set_disconnected_nodes_locs create_character_data known_ptrs
using instances
by (auto simp add: l_create_character_data_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_create_character_data_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsection ‹create\_document›
locale l_create_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_heap_is_wellformed⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs heap_is_wellformed parent_child_rel
+ l_new_document_get_disconnected_nodes
get_disconnected_nodes get_disconnected_nodes_locs
+ l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
create_document
+ l_new_document_get_child_nodes
type_wf known_ptr get_child_nodes get_child_nodes_locs
+ l_new_document
type_wf
+ l_known_ptrs
known_ptr known_ptrs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and type_wf :: "(_) heap ⇒ bool"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and set_val :: "(_) character_data_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_val_locs :: "(_) character_data_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and create_document :: "((_) heap, exception, (_) document_ptr) prog"
and known_ptrs :: "(_) heap ⇒ bool"
begin
lemma create_document_preserves_wellformedness:
assumes "heap_is_wellformed h"
and "h ⊢ create_document →⇩h h'"
and "type_wf h"
and "known_ptrs h"
shows "heap_is_wellformed h'"
proof -
obtain new_document_ptr where
new_document_ptr: "h ⊢ new_document →⇩r new_document_ptr" and
h': "h ⊢ new_document →⇩h h'"
using assms(2)
apply(simp add: create_document_def)
using new_document_ok by blast
have "new_document_ptr ∉ set |h ⊢ document_ptr_kinds_M|⇩r"
using new_document_ptr DocumentMonad.ptr_kinds_ptr_kinds_M
using new_document_ptr_not_in_heap h' by blast
then have "cast new_document_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r"
by simp
have "new_document_ptr |∉| document_ptr_kinds h"
using new_document_ptr DocumentMonad.ptr_kinds_ptr_kinds_M
using new_document_ptr_not_in_heap h' by blast
then have "cast new_document_ptr |∉| object_ptr_kinds h"
by simp
have object_ptr_kinds_eq: "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast new_document_ptr|}"
using new_document_new_ptr h' new_document_ptr by blast
then have node_ptr_kinds_eq: "node_ptr_kinds h' = node_ptr_kinds h"
apply(simp add: node_ptr_kinds_def)
by force
then have character_data_ptr_kinds_eq_h: "character_data_ptr_kinds h' = character_data_ptr_kinds h"
by(simp add: character_data_ptr_kinds_def)
have element_ptr_kinds_eq_h: "element_ptr_kinds h' = element_ptr_kinds h"
using object_ptr_kinds_eq
by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h' = document_ptr_kinds h |∪| {|new_document_ptr|}"
using object_ptr_kinds_eq
apply(auto simp add: document_ptr_kinds_def)[1]
by (metis (no_types, lifting) document_ptr_kinds_commutes document_ptr_kinds_def finsertI1 fset.map_comp)
have children_eq:
"⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_document_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h' get_child_nodes_new_document[rotated, OF new_document_ptr h']
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2: "⋀ptr'. ptr' ≠ cast new_document_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have "h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []"
using new_document_ptr h' new_document_ptr_in_heap[OF h' new_document_ptr]
new_document_is_document_ptr[OF new_document_ptr] new_document_no_child_nodes
by blast
have disconnected_nodes_eq_h:
"⋀doc_ptr disc_nodes. doc_ptr ≠ new_document_ptr
⟹ h ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes = h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads h' get_disconnected_nodes_new_document_different_pointers new_document_ptr
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by (metis(full_types) ‹⋀thesis. (⋀new_document_ptr.
⟦h ⊢ new_document →⇩r new_document_ptr; h ⊢ new_document →⇩h h'⟧ ⟹ thesis) ⟹ thesis›
local.get_disconnected_nodes_new_document_different_pointers new_document_ptr)+
then have disconnected_nodes_eq2_h: "⋀doc_ptr. doc_ptr ≠ new_document_ptr
⟹ |h ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have "h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []"
using h' local.new_document_no_disconnected_nodes new_document_ptr by blast
have "type_wf h'"
using ‹type_wf h› new_document_types_preserved h' by blast
have "acyclic (parent_child_rel h)"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def acyclic_heap_def)
also have "parent_child_rel h = parent_child_rel h'"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h'"
by (simp add: object_ptr_kinds_eq)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h' ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast new_document_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h'"
and 1: "x ∈ set |h' ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq ‹h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h'"
and 1: "x ∈ set |h' ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting) ‹h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []›
children_eq2 empty_iff empty_set image_eqI select_result_I2)
qed
finally have "a_acyclic_heap h'"
by (simp add: acyclic_heap_def)
have "a_all_ptrs_in_heap h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h'"
apply(auto simp add: a_all_ptrs_in_heap_def)[1]
using ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›
‹parent_child_rel h = parent_child_rel h'› assms(1) children_eq fset_of_list_elem
local.heap_is_wellformed_children_in_heap local.parent_child_rel_child
local.parent_child_rel_parent_in_heap node_ptr_kinds_eq
apply (metis (no_types, lifting) ‹h' ⊢ get_child_nodes (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr) →⇩r []›
children_eq2 finite_set_in finsert_iff funion_finsert_right object_ptr_kinds_eq
select_result_I2 subsetD sup_bot.right_neutral)
by (metis (no_types, lifting) ‹cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr |∉| object_ptr_kinds h›
‹h' ⊢ get_child_nodes (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr) →⇩r []›
‹h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []›
‹parent_child_rel h = parent_child_rel h'› ‹type_wf h'› assms(1) disconnected_nodes_eq_h
local.get_disconnected_nodes_ok
local.heap_is_wellformed_disc_nodes_in_heap local.parent_child_rel_child
local.parent_child_rel_parent_in_heap
node_ptr_kinds_eq returns_result_select_result select_result_I2)
have "a_distinct_lists h"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h'"
using ‹h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []›
‹h' ⊢ get_child_nodes (cast new_document_ptr) →⇩r []›
apply(auto simp add: children_eq2[symmetric] a_distinct_lists_def insort_split object_ptr_kinds_eq
document_ptr_kinds_eq_h disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1]
apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert)
apply(auto simp add: dest: distinct_concat_map_E)[1]
apply(auto simp add: dest: distinct_concat_map_E)[1]
using ‹new_document_ptr |∉| document_ptr_kinds h›
apply(auto simp add: distinct_insort dest: distinct_concat_map_E)[1]
using disconnected_nodes_eq_h
apply (metis assms(1) assms(3) disconnected_nodes_eq2_h local.get_disconnected_nodes_ok
local.heap_is_wellformed_disconnected_nodes_distinct
returns_result_select_result)
proof -
fix x :: "(_) document_ptr" and y :: "(_) document_ptr" and xa :: "(_) node_ptr"
assume a1: "x ≠ y"
assume a2: "x |∈| document_ptr_kinds h"
assume a3: "x ≠ new_document_ptr"
assume a4: "y |∈| document_ptr_kinds h"
assume a5: "y ≠ new_document_ptr"
assume a6: "distinct (concat (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h)))))"
assume a7: "xa ∈ set |h' ⊢ get_disconnected_nodes x|⇩r"
assume a8: "xa ∈ set |h' ⊢ get_disconnected_nodes y|⇩r"
have f9: "xa ∈ set |h ⊢ get_disconnected_nodes x|⇩r"
using a7 a3 disconnected_nodes_eq2_h by presburger
have f10: "xa ∈ set |h ⊢ get_disconnected_nodes y|⇩r"
using a8 a5 disconnected_nodes_eq2_h by presburger
have f11: "y ∈ set (sorted_list_of_set (fset (document_ptr_kinds h)))"
using a4 by simp
have "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h)))"
using a2 by simp
then show False
using f11 f10 f9 a6 a1 by (meson disjoint_iff_not_equal distinct_concat_map_E(1))
next
fix x xa xb
assume 0: "h' ⊢ get_disconnected_nodes new_document_ptr →⇩r []"
and 1: "h' ⊢ get_child_nodes (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr) →⇩r []"
and 2: "distinct (concat (map (λptr. |h ⊢ get_child_nodes ptr|⇩r)
(sorted_list_of_set (fset (object_ptr_kinds h)))))"
and 3: "distinct (concat (map (λdocument_ptr. |h ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h)))))"
and 4: "(⋃x∈fset (object_ptr_kinds h). set |h ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h). set |h ⊢ get_disconnected_nodes x|⇩r) = {}"
and 5: "x ∈ set |h ⊢ get_child_nodes xa|⇩r"
and 6: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
and 7: "xa |∈| object_ptr_kinds h"
and 8: "xa ≠ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr"
and 9: "xb |∈| document_ptr_kinds h"
and 10: "xb ≠ new_document_ptr"
then show "False"
by (metis ‹local.a_distinct_lists h› assms(3) disconnected_nodes_eq2_h
local.distinct_lists_no_parent local.get_disconnected_nodes_ok
returns_result_select_result)
qed
have "a_owner_document_valid h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
apply(auto simp add: a_owner_document_valid_def)[1]
by (metis ‹cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_document_ptr |∉| object_ptr_kinds h›
children_eq2 disconnected_nodes_eq2_h document_ptr_kinds_commutes finite_set_in
funion_iff node_ptr_kinds_eq object_ptr_kinds_eq)
show "heap_is_wellformed h'"
using ‹a_acyclic_heap h'› ‹a_all_ptrs_in_heap h'› ‹a_distinct_lists h'› ‹a_owner_document_valid h'›
by(simp add: heap_is_wellformed_def)
qed
end
interpretation i_create_document_wf?: l_create_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes
get_child_nodes_locs get_disconnected_nodes
get_disconnected_nodes_locs heap_is_wellformed parent_child_rel
set_val set_val_locs set_disconnected_nodes
set_disconnected_nodes_locs create_document known_ptrs
using instances
by (auto simp add: l_create_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def)
declare l_create_document_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
end
Theory Core_DOM
section‹The Core DOM›
text‹This theory is the main entry point of our formalization of the core DOM.›
theory Core_DOM
imports
"Core_DOM_Heap_WF"
begin
end
Theory Testing_Utils
theory Testing_Utils
imports Main
begin
ML ‹
val _ = Theory.setup
(Method.setup @{binding timed_code_simp}
(Scan.succeed (SIMPLE_METHOD' o (CHANGED_PROP oo (fn a => fn b => fn tac =>
let
val start = Time.now ();
val result = Code_Simp.dynamic_tac a b tac;
val t = Time.now() - start;
in
(if length (Seq.list_of result) > 0 then Output.information ("Took " ^ (Time.toString t)) else ());
result
end))))
"timed simplification with code equations");
val _ = Theory.setup
(Method.setup @{binding timed_eval}
(Scan.succeed (SIMPLE_METHOD' o (fn a => fn b => fn tac =>
let
val eval = CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 (Code_Runtime.dynamic_holds_conv a))) a) THEN'
resolve_tac a [TrueI];
val start = Time.now ();
val result = eval b tac
val t = Time.now() - start;
in
(if length (Seq.list_of result) > 0 then Output.information ("Took " ^ (Time.toString t)) else ());
result
end)))
"timed evaluation");
val _ = Theory.setup
(Method.setup @{binding timed_eval_and_code_simp}
(Scan.succeed (SIMPLE_METHOD' o (fn a => fn b => fn tac =>
let
val eval = CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 (Code_Runtime.dynamic_holds_conv a))) a) THEN'
resolve_tac a [TrueI];
val start = Time.now ();
val result = eval b tac
val t = Time.now() - start;
val start2 = Time.now ();
val result2_opt =
Timeout.apply (seconds 600.0) (fn _ => SOME (Code_Simp.dynamic_tac a b tac)) ()
handle Timeout.TIMEOUT _ => NONE;
val t2 = Time.now() - start2;
in
if length (Seq.list_of result) > 0 then (Output.information ("eval took " ^ (Time.toString t));
File.append (Path.explode "/tmp/isabellebench") (Time.toString t ^ ",")) else ();
(case result2_opt of
SOME result2 =>
(if length (Seq.list_of result2) > 0 then (Output.information ("code_simp took " ^ (Time.toString t2));
File.append (Path.explode "/tmp/isabellebench") (Time.toString t2 ^ "\n")) else ())
| NONE => (Output.information "code_simp timed out after 600s"; File.append (Path.explode "/tmp/isabellebench") (">600.000\n")));
result
end)))
"timed evaluation and simplification with code equations with file output");
›
end
Theory Core_DOM_BaseTest
section‹Common Test Setup›
text‹This theory provides the common test setup that is used by all formalized test cases.›
theory Core_DOM_BaseTest
imports
"../preliminaries/Testing_Utils"
"../Core_DOM"
begin
definition "assert_throws e p = do {
h ← get_heap;
(if (h ⊢ p →⇩e e) then return () else error AssertException)
}"
notation assert_throws ("assert'_throws'(_, _')")
definition "test p h ⟷ h ⊢ ok p"
definition field_access :: "(string ⇒ (_, (_) object_ptr option) dom_prog) ⇒ string
⇒ (_, (_) object_ptr option) dom_prog" (infix "." 80)
where
"field_access m field = m field"
definition assert_equals :: "'a ⇒ 'a ⇒ (_, unit) dom_prog"
where
"assert_equals l r = (if l = r then return () else error AssertException)"
definition assert_equals_with_message :: "'a ⇒ 'a ⇒ 'b ⇒ (_, unit) dom_prog"
where
"assert_equals_with_message l r _ = (if l = r then return () else error AssertException)"
notation assert_equals ("assert'_equals'(_, _')")
notation assert_equals_with_message ("assert'_equals'(_, _, _')")
notation assert_equals ("assert'_array'_equals'(_, _')")
notation assert_equals_with_message ("assert'_array'_equals'(_, _, _')")
definition assert_not_equals :: "'a ⇒ 'a ⇒ (_, unit) dom_prog"
where
"assert_not_equals l r = (if l ≠ r then return () else error AssertException)"
definition assert_not_equals_with_message :: "'a ⇒ 'a ⇒ 'b ⇒ (_, unit) dom_prog"
where
"assert_not_equals_with_message l r _ = (if l ≠ r then return () else error AssertException)"
notation assert_not_equals ("assert'_not'_equals'(_, _')")
notation assert_not_equals_with_message ("assert'_not'_equals'(_, _, _')")
notation assert_not_equals ("assert'_array'_not'_equals'(_, _')")
notation assert_not_equals_with_message ("assert'_array'_not'_equals'(_, _, _')")
definition removeWhiteSpaceOnlyTextNodes :: "((_) object_ptr option) ⇒ (_, unit) dom_prog"
where
"removeWhiteSpaceOnlyTextNodes _ = return ()"
subsection ‹Making the functions under test compatible with untyped languages such as JavaScript›
fun set_attribute_with_null :: "((_) object_ptr option) ⇒ attr_key ⇒ attr_value ⇒ (_, unit) dom_prog"
where
"set_attribute_with_null (Some ptr) k v = (case cast ptr of
Some element_ptr ⇒ set_attribute element_ptr k (Some v))"
fun set_attribute_with_null2 :: "((_) object_ptr option) ⇒ attr_key ⇒ attr_value option ⇒ (_, unit) dom_prog"
where
"set_attribute_with_null2 (Some ptr) k v = (case cast ptr of
Some element_ptr ⇒ set_attribute element_ptr k v)"
notation set_attribute_with_null ("_ . setAttribute'(_, _')")
notation set_attribute_with_null2 ("_ . setAttribute'(_, _')")
fun get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_with_null :: "((_) object_ptr option) ⇒ (_, (_) object_ptr option list) dom_prog"
where
"get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_with_null (Some ptr) = do {
children ← get_child_nodes ptr;
return (map (Some ∘ cast) children)
}"
notation get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_with_null ("_ . childNodes")
fun create_element_with_null :: "((_) object_ptr option) ⇒ string ⇒ (_, ((_) object_ptr option)) dom_prog"
where
"create_element_with_null (Some owner_document_obj) tag = (case cast owner_document_obj of
Some owner_document ⇒ do {
element_ptr ← create_element owner_document tag;
return (Some (cast element_ptr))})"
notation create_element_with_null ("_ . createElement'(_')")
fun create_character_data_with_null :: "((_) object_ptr option) ⇒ string ⇒ (_, ((_) object_ptr option)) dom_prog"
where
"create_character_data_with_null (Some owner_document_obj) tag = (case cast owner_document_obj of
Some owner_document ⇒ do {
character_data_ptr ← create_character_data owner_document tag;
return (Some (cast character_data_ptr))})"
notation create_character_data_with_null ("_ . createTextNode'(_')")
definition create_document_with_null :: "string ⇒ (_, ((_::linorder) object_ptr option)) dom_prog"
where
"create_document_with_null title = do {
new_document_ptr ← create_document;
html ← create_element new_document_ptr ''html'';
append_child (cast new_document_ptr) (cast html);
heap ← create_element new_document_ptr ''heap'';
append_child (cast html) (cast heap);
body ← create_element new_document_ptr ''body'';
append_child (cast html) (cast body);
return (Some (cast new_document_ptr))
}"
abbreviation "create_document_with_null2 _ _ _ ≡ create_document_with_null ''''"
notation create_document_with_null ("createDocument'(_')")
notation create_document_with_null2 ("createDocument'(_, _, _')")
fun get_element_by_id_with_null :: "((_::linorder) object_ptr option) ⇒ string ⇒ (_, ((_) object_ptr option)) dom_prog"
where
"get_element_by_id_with_null (Some ptr) id' = do {
element_ptr_opt ← get_element_by_id ptr id';
(case element_ptr_opt of
Some element_ptr ⇒ return (Some (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r element_ptr))
| None ⇒ return None)}"
| "get_element_by_id_with_null _ _ = error SegmentationFault"
notation get_element_by_id_with_null ("_ . getElementById'(_')")
fun get_elements_by_class_name_with_null ::
"((_::linorder) object_ptr option) ⇒ string ⇒ (_, ((_) object_ptr option) list) dom_prog"
where
"get_elements_by_class_name_with_null (Some ptr) class_name =
get_elements_by_class_name ptr class_name ⤜ map_M (return ∘ Some ∘ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r)"
notation get_elements_by_class_name_with_null ("_ . getElementsByClassName'(_')")
fun get_elements_by_tag_name_with_null ::
"((_::linorder) object_ptr option) ⇒ string ⇒ (_, ((_) object_ptr option) list) dom_prog"
where
"get_elements_by_tag_name_with_null (Some ptr) tag =
get_elements_by_tag_name ptr tag ⤜ map_M (return ∘ Some ∘ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r)"
notation get_elements_by_tag_name_with_null ("_ . getElementsByTagName'(_')")
fun insert_before_with_null ::
"((_::linorder) object_ptr option) ⇒ ((_) object_ptr option) ⇒ ((_) object_ptr option) ⇒
(_, ((_) object_ptr option)) dom_prog"
where
"insert_before_with_null (Some ptr) (Some child_obj) ref_child_obj_opt = (case cast child_obj of
Some child ⇒ do {
(case ref_child_obj_opt of
Some ref_child_obj ⇒ insert_before ptr child (cast ref_child_obj)
| None ⇒ insert_before ptr child None);
return (Some child_obj)}
| None ⇒ error HierarchyRequestError)"
notation insert_before_with_null ("_ . insertBefore'(_, _')")
fun append_child_with_null :: "((_::linorder) object_ptr option) ⇒ ((_) object_ptr option) ⇒
(_, unit) dom_prog"
where
"append_child_with_null (Some ptr) (Some child_obj) = (case cast child_obj of
Some child ⇒ append_child ptr child
| None ⇒ error SegmentationFault)"
notation append_child_with_null ("_ . appendChild'(_')")
fun get_body :: "((_::linorder) object_ptr option) ⇒ (_, ((_) object_ptr option)) dom_prog"
where
"get_body ptr = do {
ptrs ← ptr . getElementsByTagName(''body'');
return (hd ptrs)
}"
notation get_body ("_ . body")
fun get_document_element_with_null :: "((_::linorder) object_ptr option) ⇒
(_, ((_) object_ptr option)) dom_prog"
where
"get_document_element_with_null (Some ptr) = (case cast⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r⇩2⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r ptr of
Some document_ptr ⇒ do {
element_ptr_opt ← get_M document_ptr document_element;
return (case element_ptr_opt of
Some element_ptr ⇒ Some (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r element_ptr)
| None ⇒ None)})"
notation get_document_element_with_null ("_ . documentElement")
fun get_owner_document_with_null :: "((_::linorder) object_ptr option) ⇒
(_, ((_) object_ptr option)) dom_prog"
where
"get_owner_document_with_null (Some ptr) = (do {
document_ptr ← get_owner_document ptr;
return (Some (cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr))})"
notation get_owner_document_with_null ("_ . ownerDocument")
fun remove_with_null :: "((_::linorder) object_ptr option) ⇒ ((_) object_ptr option) ⇒
(_, ((_) object_ptr option)) dom_prog"
where
"remove_with_null (Some ptr) (Some child) = (case cast child of
Some child_node ⇒ do {
remove child_node;
return (Some child)}
| None ⇒ error NotFoundError)"
| "remove_with_null None _ = error TypeError"
| "remove_with_null _ None = error TypeError"
notation remove_with_null ("_ . remove'(')")
fun remove_child_with_null :: "((_::linorder) object_ptr option) ⇒ ((_) object_ptr option) ⇒
(_, ((_) object_ptr option)) dom_prog"
where
"remove_child_with_null (Some ptr) (Some child) = (case cast child of
Some child_node ⇒ do {
remove_child ptr child_node;
return (Some child)}
| None ⇒ error NotFoundError)"
| "remove_child_with_null None _ = error TypeError"
| "remove_child_with_null _ None = error TypeError"
notation remove_child_with_null ("_ . removeChild")
fun get_tag_name_with_null :: "((_) object_ptr option) ⇒ (_, attr_value) dom_prog"
where
"get_tag_name_with_null (Some ptr) = (case cast ptr of
Some element_ptr ⇒ get_M element_ptr tag_name)"
notation get_tag_name_with_null ("_ . tagName")
abbreviation "remove_attribute_with_null ptr k ≡ set_attribute_with_null2 ptr k None"
notation remove_attribute_with_null ("_ . removeAttribute'(_')")
fun get_attribute_with_null :: "((_) object_ptr option) ⇒ attr_key ⇒ (_, attr_value option) dom_prog"
where
"get_attribute_with_null (Some ptr) k = (case cast ptr of
Some element_ptr ⇒ get_attribute element_ptr k)"
fun get_attribute_with_null2 :: "((_) object_ptr option) ⇒ attr_key ⇒ (_, attr_value) dom_prog"
where
"get_attribute_with_null2 (Some ptr) k = (case cast ptr of
Some element_ptr ⇒ do {
a ← get_attribute element_ptr k;
return (the a)})"
notation get_attribute_with_null ("_ . getAttribute'(_')")
notation get_attribute_with_null2 ("_ . getAttribute'(_')")
fun get_parent_with_null :: "((_::linorder) object_ptr option) ⇒ (_, (_) object_ptr option) dom_prog"
where
"get_parent_with_null (Some ptr) = (case cast ptr of
Some node_ptr ⇒ get_parent node_ptr)"
notation get_parent_with_null ("_ . parentNode")
fun first_child_with_null :: "((_) object_ptr option) ⇒ (_, ((_) object_ptr option)) dom_prog"
where
"first_child_with_null (Some ptr) = do {
child_opt ← first_child ptr;
return (case child_opt of
Some child ⇒ Some (cast child)
| None ⇒ None)}"
notation first_child_with_null ("_ . firstChild")
fun adopt_node_with_null ::
"((_::linorder) object_ptr option) ⇒ ((_) object_ptr option) ⇒(_, ((_) object_ptr option)) dom_prog"
where
"adopt_node_with_null (Some ptr) (Some child) = (case cast ptr of
Some document_ptr ⇒ (case cast child of
Some child_node ⇒ do {
adopt_node document_ptr child_node;
return (Some child)}))"
notation adopt_node_with_null ("_ . adoptNode'(_')")
definition createTestTree ::
"((_::linorder) object_ptr option) ⇒ (_, (string ⇒ (_, ((_) object_ptr option)) dom_prog)) dom_prog"
where
"createTestTree ref = return (λid. get_element_by_id_with_null ref id)"
end
Theory Document_adoptNode
section‹Testing Document\_adoptNode›
text‹This theory contains the test cases for Document\_adoptNode.›
theory Document_adoptNode
imports
"Core_DOM_BaseTest"
begin
definition Document_adoptNode_heap :: heap⇩f⇩i⇩n⇩a⇩l where
"Document_adoptNode_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])),
(cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 8)] fmempty None)),
(cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6), cast (element_ptr.Ref 7)] fmempty None)),
(cast (element_ptr.Ref 3), cast (create_element_obj ''meta'' [] (fmap_of_list [(''charset'', ''utf-8'')]) None)),
(cast (element_ptr.Ref 4), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)),
(cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Document.adoptNode'')),
(cast (element_ptr.Ref 5), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''help''), (''href'', ''https://dom.spec.whatwg.org/#dom-document-adoptnode'')]) None)),
(cast (element_ptr.Ref 6), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)),
(cast (element_ptr.Ref 7), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)),
(cast (element_ptr.Ref 8), cast (create_element_obj ''body'' [cast (element_ptr.Ref 9), cast (element_ptr.Ref 10), cast (element_ptr.Ref 11)] fmempty None)),
(cast (element_ptr.Ref 9), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)),
(cast (element_ptr.Ref 10), cast (create_element_obj ''x<'' [cast (character_data_ptr.Ref 2)] fmempty None)),
(cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''x'')),
(cast (element_ptr.Ref 11), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 3)] fmempty None)),
(cast (character_data_ptr.Ref 3), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]"
definition Document_adoptNode_document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "Document_adoptNode_document = Some (cast (document_ptr.Ref 1))"
text ‹"Adopting an Element called 'x<' should work."›
lemma "test (do {
tmp0 ← Document_adoptNode_document . getElementsByTagName(''x<'');
y ← return (tmp0 ! 0);
child ← y . firstChild;
tmp1 ← y . parentNode;
tmp2 ← Document_adoptNode_document . body;
assert_equals(tmp1, tmp2);
tmp3 ← y . ownerDocument;
assert_equals(tmp3, Document_adoptNode_document);
tmp4 ← Document_adoptNode_document . adoptNode(y);
assert_equals(tmp4, y);
tmp5 ← y . parentNode;
assert_equals(tmp5, None);
tmp6 ← y . firstChild;
assert_equals(tmp6, child);
tmp7 ← y . ownerDocument;
assert_equals(tmp7, Document_adoptNode_document);
tmp8 ← child . ownerDocument;
assert_equals(tmp8, Document_adoptNode_document);
doc ← createDocument(None, None, None);
tmp9 ← doc . adoptNode(y);
assert_equals(tmp9, y);
tmp10 ← y . parentNode;
assert_equals(tmp10, None);
tmp11 ← y . firstChild;
assert_equals(tmp11, child);
tmp12 ← y . ownerDocument;
assert_equals(tmp12, doc);
tmp13 ← child . ownerDocument;
assert_equals(tmp13, doc)
}) Document_adoptNode_heap"
by eval
text ‹"Adopting an Element called ':good:times:' should work."›
lemma "test (do {
x ← Document_adoptNode_document . createElement('':good:times:'');
tmp0 ← Document_adoptNode_document . adoptNode(x);
assert_equals(tmp0, x);
doc ← createDocument(None, None, None);
tmp1 ← doc . adoptNode(x);
assert_equals(tmp1, x);
tmp2 ← x . parentNode;
assert_equals(tmp2, None);
tmp3 ← x . ownerDocument;
assert_equals(tmp3, doc)
}) Document_adoptNode_heap"
by eval
end
Theory Document_getElementById
section‹Testing Document\_getElementById›
text‹This theory contains the test cases for Document\_getElementById.›
theory Document_getElementById
imports
"Core_DOM_BaseTest"
begin
definition Document_getElementById_heap :: heap⇩f⇩i⇩n⇩a⇩l where
"Document_getElementById_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])),
(cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 9)] fmempty None)),
(cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6), cast (element_ptr.Ref 7), cast (element_ptr.Ref 8)] fmempty None)),
(cast (element_ptr.Ref 3), cast (create_element_obj ''meta'' [] (fmap_of_list [(''charset'', ''utf-8'')]) None)),
(cast (element_ptr.Ref 4), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)),
(cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Document.getElementById'')),
(cast (element_ptr.Ref 5), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''author''), (''title'', ''Tetsuharu OHZEKI''), (''href'', ''mailto:saneyuki.snyk@gmail.com'')]) None)),
(cast (element_ptr.Ref 6), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''help''), (''href'', ''https://dom.spec.whatwg.org/#dom-document-getelementbyid'')]) None)),
(cast (element_ptr.Ref 7), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)),
(cast (element_ptr.Ref 8), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)),
(cast (element_ptr.Ref 9), cast (create_element_obj ''body'' [cast (element_ptr.Ref 10), cast (element_ptr.Ref 11), cast (element_ptr.Ref 12), cast (element_ptr.Ref 13), cast (element_ptr.Ref 16), cast (element_ptr.Ref 19)] fmempty None)),
(cast (element_ptr.Ref 10), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)),
(cast (element_ptr.Ref 11), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', '''')]) None)),
(cast (element_ptr.Ref 12), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''test1'')]) None)),
(cast (element_ptr.Ref 13), cast (create_element_obj ''div'' [cast (element_ptr.Ref 14), cast (element_ptr.Ref 15)] (fmap_of_list [(''id'', ''test5''), (''data-name'', ''1st'')]) None)),
(cast (element_ptr.Ref 14), cast (create_element_obj ''p'' [cast (character_data_ptr.Ref 2)] (fmap_of_list [(''id'', ''test5''), (''data-name'', ''2nd'')]) None)),
(cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''P'')),
(cast (element_ptr.Ref 15), cast (create_element_obj ''input'' [] (fmap_of_list [(''id'', ''test5''), (''type'', ''submit''), (''value'', ''Submit''), (''data-name'', ''3rd'')]) None)),
(cast (element_ptr.Ref 16), cast (create_element_obj ''div'' [cast (element_ptr.Ref 17)] (fmap_of_list [(''id'', ''outer'')]) None)),
(cast (element_ptr.Ref 17), cast (create_element_obj ''div'' [cast (element_ptr.Ref 18)] (fmap_of_list [(''id'', ''middle'')]) None)),
(cast (element_ptr.Ref 18), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''inner'')]) None)),
(cast (element_ptr.Ref 19), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 3)] fmempty None)),
(cast (character_data_ptr.Ref 3), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]"
definition Document_getElementById_document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "Document_getElementById_document = Some (cast (document_ptr.Ref 1))"
text ‹"Document.getElementById with a script-inserted element"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test2'';
test ← Document_getElementById_document . createElement(''div'');
test . setAttribute(''id'', TEST_ID);
gBody . appendChild(test);
result ← Document_getElementById_document . getElementById(TEST_ID);
assert_not_equals(result, None, ''should not be null.'');
tmp0 ← result . tagName;
assert_equals(tmp0, ''div'', ''should have appended element's tag name'');
gBody . removeChild(test);
removed ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(removed, None, ''should not get removed element.'')
}) Document_getElementById_heap"
by eval
text ‹"update `id` attribute via setAttribute/removeAttribute"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test3'';
test ← Document_getElementById_document . createElement(''div'');
test . setAttribute(''id'', TEST_ID);
gBody . appendChild(test);
UPDATED_ID ← return ''test3-updated'';
test . setAttribute(''id'', UPDATED_ID);
e ← Document_getElementById_document . getElementById(UPDATED_ID);
assert_equals(e, test, ''should get the element with id.'');
old ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(old, None, ''shouldn't get the element by the old id.'');
test . removeAttribute(''id'');
e2 ← Document_getElementById_document . getElementById(UPDATED_ID);
assert_equals(e2, None, ''should return null when the passed id is none in document.'')
}) Document_getElementById_heap"
by eval
text ‹"Ensure that the id attribute only affects elements present in a document"›
lemma "test (do {
TEST_ID ← return ''test4-should-not-exist'';
e ← Document_getElementById_document . createElement(''div'');
e . setAttribute(''id'', TEST_ID);
tmp0 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp0, None, ''should be null'');
tmp1 ← Document_getElementById_document . body;
tmp1 . appendChild(e);
tmp2 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp2, e, ''should be the appended element'')
}) Document_getElementById_heap"
by eval
text ‹"in tree order, within the context object's tree"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test5'';
target ← Document_getElementById_document . getElementById(TEST_ID);
assert_not_equals(target, None, ''should not be null'');
tmp0 ← target . getAttribute(''data-name'');
assert_equals(tmp0, ''1st'', ''should return the 1st'');
element4 ← Document_getElementById_document . createElement(''div'');
element4 . setAttribute(''id'', TEST_ID);
element4 . setAttribute(''data-name'', ''4th'');
gBody . appendChild(element4);
target2 ← Document_getElementById_document . getElementById(TEST_ID);
assert_not_equals(target2, None, ''should not be null'');
tmp1 ← target2 . getAttribute(''data-name'');
assert_equals(tmp1, ''1st'', ''should be the 1st'');
tmp2 ← target2 . parentNode;
tmp2 . removeChild(target2);
target3 ← Document_getElementById_document . getElementById(TEST_ID);
assert_not_equals(target3, None, ''should not be null'');
tmp3 ← target3 . getAttribute(''data-name'');
assert_equals(tmp3, ''4th'', ''should be the 4th'')
}) Document_getElementById_heap"
by eval
text ‹"Modern browsers optimize this method with using internal id cache. This test checks that their optimization should effect only append to `Document`, not append to `Node`."›
lemma "test (do {
TEST_ID ← return ''test6'';
s ← Document_getElementById_document . createElement(''div'');
s . setAttribute(''id'', TEST_ID);
tmp0 ← Document_getElementById_document . createElement(''div'');
tmp0 . appendChild(s);
tmp1 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp1, None, ''should be null'')
}) Document_getElementById_heap"
by eval
text ‹"changing attribute's value via `Attr` gotten from `Element.attribute`."›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test7'';
element ← Document_getElementById_document . createElement(''div'');
element . setAttribute(''id'', TEST_ID);
gBody . appendChild(element);
target ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(target, element, ''should return the element before changing the value'');
element . setAttribute(''id'', (TEST_ID @ ''-updated''));
target2 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(target2, None, ''should return null after updated id via Attr.value'');
target3 ← Document_getElementById_document . getElementById((TEST_ID @ ''-updated''));
assert_equals(target3, element, ''should be equal to the updated element.'')
}) Document_getElementById_heap"
by eval
text ‹"update `id` attribute via element.id"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test12'';
test ← Document_getElementById_document . createElement(''div'');
test . setAttribute(''id'', TEST_ID);
gBody . appendChild(test);
UPDATED_ID ← return (TEST_ID @ ''-updated'');
test . setAttribute(''id'', UPDATED_ID);
e ← Document_getElementById_document . getElementById(UPDATED_ID);
assert_equals(e, test, ''should get the element with id.'');
old ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(old, None, ''shouldn't get the element by the old id.'');
test . setAttribute(''id'', '''');
e2 ← Document_getElementById_document . getElementById(UPDATED_ID);
assert_equals(e2, None, ''should return null when the passed id is none in document.'')
}) Document_getElementById_heap"
by eval
text ‹"where insertion order and tree order don't match"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test13'';
container ← Document_getElementById_document . createElement(''div'');
container . setAttribute(''id'', (TEST_ID @ ''-fixture''));
gBody . appendChild(container);
element1 ← Document_getElementById_document . createElement(''div'');
element1 . setAttribute(''id'', TEST_ID);
element2 ← Document_getElementById_document . createElement(''div'');
element2 . setAttribute(''id'', TEST_ID);
element3 ← Document_getElementById_document . createElement(''div'');
element3 . setAttribute(''id'', TEST_ID);
element4 ← Document_getElementById_document . createElement(''div'');
element4 . setAttribute(''id'', TEST_ID);
container . appendChild(element2);
container . appendChild(element4);
container . insertBefore(element3, element4);
container . insertBefore(element1, element2);
test ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(test, element1, ''should return 1st element'');
container . removeChild(element1);
test ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(test, element2, ''should return 2nd element'');
container . removeChild(element2);
test ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(test, element3, ''should return 3rd element'');
container . removeChild(element3);
test ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(test, element4, ''should return 4th element'');
container . removeChild(element4)
}) Document_getElementById_heap"
by eval
text ‹"Inserting an id by inserting its parent node"›
lemma "test (do {
gBody ← Document_getElementById_document . body;
TEST_ID ← return ''test14'';
a ← Document_getElementById_document . createElement(''a'');
b ← Document_getElementById_document . createElement(''b'');
a . appendChild(b);
b . setAttribute(''id'', TEST_ID);
tmp0 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp0, None);
gBody . appendChild(a);
tmp1 ← Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp1, b)
}) Document_getElementById_heap"
by eval
text ‹"Document.getElementById must not return nodes not present in document"›
lemma "test (do {
TEST_ID ← return ''test15'';
outer ← Document_getElementById_document . getElementById(''outer'');
middle ← Document_getElementById_document . getElementById(''middle'');
inner ← Document_getElementById_document . getElementById(''inner'');
tmp0 ← Document_getElementById_document . getElementById(''middle'');
outer . removeChild(tmp0);
new_el ← Document_getElementById_document . createElement(''h1'');
new_el . setAttribute(''id'', ''heading'');
inner . appendChild(new_el);
tmp1 ← Document_getElementById_document . getElementById(''heading'');
assert_equals(tmp1, None)
}) Document_getElementById_heap"
by eval
end
Theory Node_insertBefore
section‹Testing Node\_insertBefore›
text‹This theory contains the test cases for Node\_insertBefore.›
theory Node_insertBefore
imports
"Core_DOM_BaseTest"
begin
definition Node_insertBefore_heap :: heap⇩f⇩i⇩n⇩a⇩l where
"Node_insertBefore_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])),
(cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 6)] fmempty None)),
(cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5)] fmempty None)),
(cast (element_ptr.Ref 3), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)),
(cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Node.insertBefore'')),
(cast (element_ptr.Ref 4), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)),
(cast (element_ptr.Ref 5), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)),
(cast (element_ptr.Ref 6), cast (create_element_obj ''body'' [cast (element_ptr.Ref 7), cast (element_ptr.Ref 8)] fmempty None)),
(cast (element_ptr.Ref 7), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)),
(cast (element_ptr.Ref 8), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 2)] fmempty None)),
(cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]"
definition Node_insertBefore_document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "Node_insertBefore_document = Some (cast (document_ptr.Ref 1))"
text ‹"Calling insertBefore an a leaf node Text must throw HIERARCHY\_REQUEST\_ERR."›
lemma "test (do {
node ← Node_insertBefore_document . createTextNode(''Foo'');
tmp0 ← Node_insertBefore_document . createTextNode(''fail'');
assert_throws(HierarchyRequestError, node . insertBefore(tmp0, None))
}) Node_insertBefore_heap"
by eval
text ‹"Calling insertBefore with an inclusive ancestor of the context object must throw HIERARCHY\_REQUEST\_ERR."›
lemma "test (do {
tmp1 ← Node_insertBefore_document . body;
tmp2 ← Node_insertBefore_document . getElementById(''log'');
tmp0 ← Node_insertBefore_document . body;
assert_throws(HierarchyRequestError, tmp0 . insertBefore(tmp1, tmp2));
tmp4 ← Node_insertBefore_document . documentElement;
tmp5 ← Node_insertBefore_document . getElementById(''log'');
tmp3 ← Node_insertBefore_document . body;
assert_throws(HierarchyRequestError, tmp3 . insertBefore(tmp4, tmp5))
}) Node_insertBefore_heap"
by eval
text ‹"Calling insertBefore with a reference child whose parent is not the context node must throw a NotFoundError."›
lemma "test (do {
a ← Node_insertBefore_document . createElement(''div'');
b ← Node_insertBefore_document . createElement(''div'');
c ← Node_insertBefore_document . createElement(''div'');
assert_throws(NotFoundError, a . insertBefore(b, c))
}) Node_insertBefore_heap"
by eval
text ‹"If the context node is a document, inserting a document or text node should throw a HierarchyRequestError."›
lemma "test (do {
doc ← createDocument(''title'');
doc2 ← createDocument(''title2'');
tmp0 ← doc . documentElement;
assert_throws(HierarchyRequestError, doc . insertBefore(doc2, tmp0));
tmp1 ← doc . createTextNode(''text'');
tmp2 ← doc . documentElement;
assert_throws(HierarchyRequestError, doc . insertBefore(tmp1, tmp2))
}) Node_insertBefore_heap"
by eval
text ‹"Inserting a node before itself should not move the node"›
lemma "test (do {
a ← Node_insertBefore_document . createElement(''div'');
b ← Node_insertBefore_document . createElement(''div'');
c ← Node_insertBefore_document . createElement(''div'');
a . appendChild(b);
a . appendChild(c);
tmp0 ← a . childNodes;
assert_array_equals(tmp0, [b, c]);
tmp1 ← a . insertBefore(b, b);
assert_equals(tmp1, b);
tmp2 ← a . childNodes;
assert_array_equals(tmp2, [b, c]);
tmp3 ← a . insertBefore(c, c);
assert_equals(tmp3, c);
tmp4 ← a . childNodes;
assert_array_equals(tmp4, [b, c])
}) Node_insertBefore_heap"
by eval
end
Theory Node_removeChild
section‹Testing Node\_removeChild›
text‹This theory contains the test cases for Node\_removeChild.›
theory Node_removeChild
imports
"Core_DOM_BaseTest"
begin
definition Node_removeChild_heap :: heap⇩f⇩i⇩n⇩a⇩l where
"Node_removeChild_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])),
(cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 7)] fmempty None)),
(cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6)] fmempty None)),
(cast (element_ptr.Ref 3), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)),
(cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Node.removeChild'')),
(cast (element_ptr.Ref 4), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)),
(cast (element_ptr.Ref 5), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)),
(cast (element_ptr.Ref 6), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''creators.js'')]) None)),
(cast (element_ptr.Ref 7), cast (create_element_obj ''body'' [cast (element_ptr.Ref 8), cast (element_ptr.Ref 9), cast (element_ptr.Ref 10)] fmempty None)),
(cast (element_ptr.Ref 8), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)),
(cast (element_ptr.Ref 9), cast (create_element_obj ''iframe'' [] (fmap_of_list [(''src'', ''about:blank'')]) None)),
(cast (element_ptr.Ref 10), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 2)] fmempty None)),
(cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]"
definition Node_removeChild_document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "Node_removeChild_document = Some (cast (document_ptr.Ref 1))"
text ‹"Passing a detached Element to removeChild should not affect it."›
lemma "test (do {
doc ← return Node_removeChild_document;
s ← doc . createElement(''div'');
tmp0 ← s . ownerDocument;
assert_equals(tmp0, doc);
tmp1 ← Node_removeChild_document . body;
assert_throws(NotFoundError, tmp1 . removeChild(s));
tmp2 ← s . ownerDocument;
assert_equals(tmp2, doc)
}) Node_removeChild_heap"
by eval
text ‹"Passing a non-detached Element to removeChild should not affect it."›
lemma "test (do {
doc ← return Node_removeChild_document;
s ← doc . createElement(''div'');
tmp0 ← doc . documentElement;
tmp0 . appendChild(s);
tmp1 ← s . ownerDocument;
assert_equals(tmp1, doc);
tmp2 ← Node_removeChild_document . body;
assert_throws(NotFoundError, tmp2 . removeChild(s));
tmp3 ← s . ownerDocument;
assert_equals(tmp3, doc)
}) Node_removeChild_heap"
by eval
text ‹"Calling removeChild on an Element with no children should throw NOT\_FOUND\_ERR."›
lemma "test (do {
doc ← return Node_removeChild_document;
s ← doc . createElement(''div'');
tmp0 ← doc . body;
tmp0 . appendChild(s);
tmp1 ← s . ownerDocument;
assert_equals(tmp1, doc);
assert_throws(NotFoundError, s . removeChild(doc))
}) Node_removeChild_heap"
by eval
text ‹"Passing a detached Element to removeChild should not affect it."›
lemma "test (do {
doc ← createDocument('''');
s ← doc . createElement(''div'');
tmp0 ← s . ownerDocument;
assert_equals(tmp0, doc);
tmp1 ← Node_removeChild_document . body;
assert_throws(NotFoundError, tmp1 . removeChild(s));
tmp2 ← s . ownerDocument;
assert_equals(tmp2, doc)
}) Node_removeChild_heap"
by eval
text ‹"Passing a non-detached Element to removeChild should not affect it."›
lemma "test (do {
doc ← createDocument('''');
s ← doc . createElement(''div'');
tmp0 ← doc . documentElement;
tmp0 . appendChild(s);
tmp1 ← s . ownerDocument;
assert_equals(tmp1, doc);
tmp2 ← Node_removeChild_document . body;
assert_throws(NotFoundError, tmp2 . removeChild(s));
tmp3 ← s . ownerDocument;
assert_equals(tmp3, doc)
}) Node_removeChild_heap"
by eval
text ‹"Calling removeChild on an Element with no children should throw NOT\_FOUND\_ERR."›
lemma "test (do {
doc ← createDocument('''');
s ← doc . createElement(''div'');
tmp0 ← doc . body;
tmp0 . appendChild(s);
tmp1 ← s . ownerDocument;
assert_equals(tmp1, doc);
assert_throws(NotFoundError, s . removeChild(doc))
}) Node_removeChild_heap"
by eval
text ‹"Passing a value that is not a Node reference to removeChild should throw TypeError."›
lemma "test (do {
tmp0 ← Node_removeChild_document . body;
assert_throws(TypeError, tmp0 . removeChild(None))
}) Node_removeChild_heap"
by eval
end